The philosopher David Chalmers has speculated on a number of ways in which quantum mechanics might relate to consciousness.
"One possibility is that instead of postulating novel properties, physics might end up appealing to consciousness itself, in the way that some theorists hold that quantum mechanics does." 
"The collapse dynamics leaves a door wide open for an interactionist interpretation". 
"The most promising version of such an interpretation allows conscious states to be correlated with the total quantum state of a system, with the extra constraint that conscious states (unlike physical states) can never be superposed. In a conscious physical system such as a brain, the physical and phenomenal states of the system will be correlated in a (nonsuperposed) quantum state. Upon observation of a superposed external system, Schrödinger evolution at the moment of observation would cause the observed system to become correlated with the brain, yielding a resulting superposition of brain states and so (by psychophysical correlation) a superposition of conscious states. But such a superposition cannot occur, so one of the potential resulting conscious states is somehow selected (presumably by a nondeterministic dynamic principle at the phenomenal level). The result is that (by psychophysical correlation) a definite brain state and a definite state of the observed object are also selected". 
"If physics is supposed to rule out interactionism, then careful attention to the detail of physical theory is required". 
In The Emperor's New Mind and Shadows of the Mind, Penrose argues that
Humans have abilities, particularly mathematical ones, that no algorithmic computer (specifically Turing machine) could have, because computers are limited by Godel's incompleteness Theorem. In other words, he believes humans are hypercomputers. (The argument was originally due to John Lucas.)
Godel demonstrated that with any set of axioms, it was possile to produce a statement that was obviously true, but could not be proved by the axioms. The theorem enjoys general acceptance in the mathematical community.
Penrose, however, built a further and highly controversial argument on this theorem. He argued that the theorem showed that the brain had the ability to go beyond what can be demonstrated by mathematical axioms, and therefore there is something within the functioning of the brain that is not based on an algorithm (a system of calculations). A computer is just a system of algorithms, and Penrose claimed that Godel's theorem demonstrated that brains could perform functions that no computer could perform.
These assertions have been vigorously contested by many critics and notably by the philosophers Churchland and Grush.
2. This would require some new physics. Penrose postulates that the currently unknown process underlying quantum collapse supplies the non-algorithmic element.
The random choice of, for instance, the position of a particle, which is involved in the collapse of the wave function was the only physical process that Penrose could find, which was not based on an algorithm. However, randomness was not a promising basis for the quality of mathematical judgement highlighted by his Godel theorem argument.
But Penrose went on to propose that when the wave function did not collapse as a result of a measurement or decoherence in the environment, there could be an alternative form of wave function collapse, which he called objective reduction(OR). In this, each quantum superposition has its own space time geometry. When these become separated by more then the Planck length, they are effected by gravity, become unstable and collapse. OR is strikingly different both from the traditional orthodoxy of Neils Bohr's Copenhagen interpretation of quantum theory and from some more modern theories which avoid wave function collapse altogether such as 'many worlds' or some forms of decoherence theory.
Penrose further proposes that OR is neither random nor governed by an algorithm, but is 'non-computational', selecting information embedded in the fundamental level of space time geometry.
3. Collapse requires a coherent superposed state to work on. Penrose borrows Stuart Hameroff's proposal about microtubules to supply this.
Initially, Penrose had lacked any detailed proposals for how OR could occur in the brain. Later on cooperation with Stuart Hameroff  supplied this side of the theory. Microtubules were central to Hameroff's proposals. These are the core element of the cytoskeleton, which provides a supportive structure and performs various functions in body cells. In additions to these functions, it was now proposed that the microtubules could support macroscopic quantum features known as Bose condensates. It was also suggested that these condensates could link with other neurons via gap junctions. This is claimed to permit quantum coherence to extend over a large area of the brain. It is suggested that when one of these areas of quantum coherence collapses, there is an instance of consciousness, and the brain has access to a non-computational process embedded in the fundamental level of space time geometry.
At the same time, it was postulated that conventional synaptic activity influences and is influenced by the activity in the microtubules. This part of the process is referred to as 'orchestration' hence the theory is called Orchestrated Objective Reduction or more commonly Orch OR.
Hameroff's proposals like those of Penrose attracted much criticism. However the most cogent attach on Orch OR and quantum mind theories in general was the view that conditions in the brain would lead to any quantum coherence decohering too quickly for it to be relevant to neural processes. This general criticism is discussed in the Science section below.
(Penrose is not interested in explaining phenomenal consciousness, qualia, generally regarded as the most mysterious feature of consciousness, but instead focuses mainly on the cognitive powers of mathematicians.)
The theory has been much criticised   
Physically, Stapp's approach is aligned with objective collapse theory, in that the deterministic evolution of the wave function, and its indeterministic collapse are seen as two real and ontologically distinct phenomena. Collapse events occurring within the brain — the mind's observation or measurement of the brain — are particularly important. Since Stapp sees collapse as a mental process and the deterministic evolution of brain states as physical, his approach is philosophically aligned with interactionist dualism. The process by which collapse selects and actuality from a set of possibilities is seen by Stapp as a literally a process of choice, and not merely a random dice-throw. His approach has implications with regard to time. Since the future depends on decisions in the present, it is not pre-existing, as in the block universe theory; rather there is an evolving universe in which subjects participate, as in Whitehead's metaphysics. 
Stapp envisages consciousness as exercising top-level control over neural excitation in the brain. Quantum brain events are suggested to occur at the whole brain level, and are seen as being selected from the large-scale excitation of the brain. The neural excitations are viewed as a code, and each conscious experience as a selection from this code. The brain, in this theory, is proposed to be a self-programming computer with a self-sustaining input from memory, which is itself a code derived from previous experience. This process results in a number of probabilities from which consciousness has to select. The conscious act is a selection of a piece of top-level code, which then exercises ongoing control over the flow of neural excitation. This process refers to the top levels of brain activity involved with information gathering, planning and the monitoring of the execution of plans. Conscious events are proposed to be capable of grasping a whole pattern of activity, thus accounting for the unity of consciousness, and providing a solution to the 'binding problem'.
Stapp's version of the conscious brain is proposed to be a system that is internally determined in a way that cannot be represented outside the system, whereas for the rest of the physical universe an external representation plus a knowledge of the laws of physics allows an accurate prediction of future events.
Stapp proposes that the proof of his theory requires the identification of the neurons that provide the top-level code and also the process by which memory is turned into additional top-level code.
David Bohm took the view that quantum theory and relativity contradicted one another, and that this contradiction implied that there existed a more fundamental level in the physical universe. He claimed that both quantum theory and relativity pointed towards this deeper theory. This more fundamental level was supposed to represent an undivided wholeness and an implicate order, from which arose the explicate order of the universe as we experience it.
Bohm's implicate order applies both to matter and consciounsess, and he proposed that it could explain the relationship between them. Mind and matter are here seen as projections into our explicate order from the underlying reality of the implicate order. Bohm claims that when we look at the matter in space, we can see nothing in these concepts that helps us to understand consciousness.
In Bohm's scheme there is a fundamental level where consciousness is not distinct from matter. Bohm's view of consciousness is connected to Karl Pribram's holographic conception of the brain  . Pribram regards sight and the other senses as lenses without which the other senses would appear as a hologram. Pribram proposes that information is recorded all over the brain, and that it is enfolded into a whole, similar to a hologram. It is suggested that memories are connected by association and manipulated by logical thought. If the brain is also receiving sensory input all these are proposed to unite in overall experience or consciousness.
In trying to describe the nature of consciousness, Bohm discusses the experience of listening to music. He thinks that the feeling of movement and change that make up our experience of music derives from both the immediate past and the present being held in the brain together, with the notes from the past seen as transformations rather than memories. The notes that were implicate in the immediate past are seen as becoming explicate in the present. Bohm compares this to consciousness emerging from the implicate order.
Bohm sees the movement, change or flow and also the coherence of experiences such as listening to music as a manifestation of the implicate order. He claims to derive evidence for this from the work of Piaget in studying infants. He claims that these studies show that young children have to learn about time and space, because they are part of the explicate order, but have a 'hard-wired' understanding of movement because it is part of the implicate order. He compares this 'hard-wiring' to Chomsky's theory that grammar is 'hard-wired' into young human brains.
In his writings, Bohm never proposed any specific brain mechanism by which his implicate order could emerge in a way that was relevant to consciousness. However, recent papers by physicist, Gustav Bernroider, have indicated that he thinks that Bohm's implicate-explicate structure can account for the relationship between neural processes and consciousness. In a paper published in 2005 Bernroider elaborated his proposals for the physical basis of this process. The main thrust of his paper was the argument that quantum coherence may be sustained in ion channels for long enough to be relevant for neural processes and the channels could be entangled with surrounding lipids and proteins and with other channels in the same membrane. Ion channels regulate the electrical potential across the axon membrane and thus play a central role in the brain's information processing.
Bernroider bases his work on recent studies of the potassium (K+)ion channel in its closed state and draws particularly on the atomic-level spectroscopy work of the MacKinnon group . The ion channels have a filter region which allows in K+ ions and bars other ions. These studies show that the filter region has a framework of five sets of four oxygen atoms, which are part of the carboxyl group of amino-acid molecules in the surrounding protein. These are referred to as binding pockets. Two K+ ions are trapped in the selection filter of the closed ion channel. Each of these ions is electostatically bound to two sets of oxygen atoms or binding pockets, involving eight oxygen atoms in total. Both ions in the channel oscillate between two configurations.
Bernroider uses this recently revealed structure to speculate about the possibility of quantum coherence in the ion channels. Bernroider and co-author Sisir Roy's calculations suggested to them that the behaviour of the ions in the K channel could only be understood at the quantum level. Taking this as their starting point, they then ask whether the structure of the ion channel can be related to logic states. Further calculations lead them to suggest that the K+ ions and the oxygen atoms of the binding pockets are two quantum-entangled sub-systems, which they then equate to a quantum computational mapping. The ions that are destined to be expelled from the channel are proposed to encode information about the state of the oxygen atoms. It is further proposed the separate ion channels could be quantum entangled with one another
Quantum Brain Dynamics
The ideas behind QBD derived originally from the physicists, Hiroomi Umezawa, and Herbert Frohlich in the 1960s. In recent decades these ideas have been elaborated and given greater prominence by younger physicists such as Mari Jibu, Kunio Yasue and Giuseppe Vitiello.In quantum brain dynamics(QBD)the electrical dipoles of the water molecules that constitute 70% of the brain are proposed to constitute a quantum field, known here as the cortical field. The quanta of this field are described as corticons. In the theory, this field interacts with quantum coherent waves generated by biomolecules in the neurons and propagating along the neuronal network.
Frohlich is the source of the idea that quantum coherent waves could be generated in the neuronal network. Frohlich argued that it was not clear how order could be sustained in living systems given the disruptive influence of the fluctuations in biochemical processes. He viewed the electric potential across the neuron membrane as the observable feature of some form of underlying quantum order. His studies claimed to show that with an oscillating charge in a thermal bath, large numbers of quanta may condense into a single state known as a Bose condensate. This state allows long-range correlation amongst the dipoles involved. Further to this, biomolecules were proposed to line up along actin filaments (part of the cytoskeleton) and dipole oscillations propagate along the filaments as quantum coherent waves. This now has some experimental support in the form of confirmation that biomolecules with high electric dipole moment have been shown to have a periodic oscillation. Vitiello also argues that the ordered chains of chemical reactions on which biological tissues depend would collapse without some form of quantum ordering, which in QBD is described by quantum field theory rather than quantum mechanics.
Vitiello provides citations, which are claimed to support his view of biological tissue. These include studies of radiation effect on cell growth,response to external stimuli, non-linear tunnelling,coherent nuclear motion in membrane proteins,optical coherence in biological systems, energy transfer via solitons and coherent excitations.
QBD proposes that the cortical field not only interacts with, but also to a good extent controls the neuronal network. It suggests that biomolecular waves propagate along the actin filaments in the area of the cell membranes and dendritic spines. The waves derive energy from ATP molecules stored in the cell membrane and control the ion channels, which in turn regulate the flow of signals to the synapses. Vitiello claims that QBD does not require quantum oscillations to last as long as the actual time to decoherence.
The proponents of QBD differ somewhat as the exact way in which it produces consciousness. Jibu and Yasue think that the interaction between the energy quanta of the cortical field and the biomolecular waves of the neuronal network, particularly the dendritic part of the network, is what produces consciousness. On the other hand, Vitiello thinks that the quantum states involved in QBD produce two poles, a subjective representation of the external world and a self. This self opens itself to the representation of the external world. Consciousness is, in this theory, not in either the self or the external representation, but between the two in the opening of one to the other.