The Stanford Online Encyclopedia of Philosophy has a substantially revised version of its entry on Quantum Approaches to Consciousness, and it's a pretty decent introduction to the subject.
The author, Harald Atmanspacher, begins by pointing out that because correlation doesn't entail causation, the correlation between particular mental experiences and particular areas of brain activity (as revealed in magnetic resonance imagery, for example), doesn't entail that brain activity causes those mental experiences.
This observation provides the point-of-entry for so-called dual aspect theories of the mind-brain relationship, which suggest that the mind and the brain are two different aspects of some underlying, unified reality, in contrast to the notion that the mind can be reduced to the brain. One analogy often used in this context is the relationship between electricity and magnetism: these two phenomena are merely different aspects of a unified entity, the electromagnetic field, and although there are very strong correlations between electricity and magnetism, the existence of those correlations does not entail that electricity can be reduced to magnetism or vice versa.
David Bohm's account of the mind-matter relationship, for example, falls into the category of dual aspect theories, claiming as it does the existence of an 'implicate order which unfolds into the different explicate domains of the mental and the material'.
The basic idea of these dual aspect theories is a good one, but those who espouse such approaches need to appreciate that the relationship between the mind and the brain is not one characterised merely by correlations. Rather, there is a much stronger coarse-graining relationship, in which mental states correspond to entire classes of brain states. If a brain-state is altered on a nanoscopic level, it doesn't change the corresponding mental state; each mental state corresponds to an entire class of nanoscopically distinct brain-states. It is this coarse-graining relationship which entails that the mind supervenes upon the brain, and not vice-versa. Arguably, it is precisely this asymmetry which suggests that the mind reduces to, or emerges from the brain, rather than the mind and the brain being related by a duality transformation.
Nevertheless, let's try to imagine how a dual-aspect theory of the mind-brain relationship might work. For a start, we'd need to characterise both the mind and the brain in formal terms, just as we do with electicity and magnetism. We might, for example, characterise the brain as a neural network, an abstraction from the network of nerve cells and synapses in a biological brain. A formal theory of the mind doesn't exist as yet, but the best nascent candidate is perhaps the Representational Theory of the Mind (RTM). So let's just briefly describe these two theories.
A neural network consists of a set of nodes, and a set of connections between the nodes. The nodes in a neural network possess activation levels, the connections between nodes possess weights, and the nodes have numerical rules for calculating their next activation level from (i) the previous activation level, and (ii) the weighted inputs from other nodes.
The RTM, meanwhile, attempts to provide an account of intentional mental states. These are states, such as beliefs and desires, in which the attention of the mind is directed towards something, called the 'content' of the intentional state. Many advocates of the RTM claim that the mental representations which provide the content of intentional states, possess an internal structure. They hold that this internal system of representation has a set of symbols, a syntax, and a semantics, collectively termed the language of thought. There are rules for composing the symbols into expressions, propositions, and mental images, hence the content of an intentional state can be said to possess a symbol structure. The RTM considers mental processes such as thinking, reasoning and imagining to be sequences of intentional mental states.
A dual aspect theory of the mind-brain relationship would need to find an underlying structure which incorporates both the structure of neural networks and the structure described by the RTM, and relates the two by a duality transformation.
Neural networks, of course, are part of classical physics, so whether quantum theory is actually relevant to the mind-brain relationship is another matter entirely...