A usual response to speculating about any sort of connection between quantum physics and consciousness is that quantum effects do not have any meaningful influence at the scales where we expect the answers to the mysteries of consciousness lie.
In a sense, this is true (or may be true: recent discoveries regarding photosynthesis and avian magnetoception have at least open the doors towards speculating a role for quantum effects in biology), but besides the point: because additionally to some neat new phenomena to play with, a new physical theory also invites new metaphysics, new ways of thinking about what may be physically possible—new tools for explaining stuff.
And indeed, quantum physics has already amassed an impressive track record in this regard: for one, the stability and extension of ordinary matter is an unexplained posit of classical physics—but it finds an explanation in quantum theory: the quantization of angular momentum ensures stable electron configurations; additionally, the Pauli exclusion principle forbids everything to just go on top of every other thing.
So, having already explained the central property of Descartes’ res extensa—its extendedness—even though there are no significant quantum effects on the level of tables and chairs, one might at least make a poetic case that it’d be only right for QM to do the same to his res cogitans. But is there more than poetry to it? Well, nobody knows, as of yet. I think one interesting direction is that quantum mechanics has shown us that there are more ways that one can combine properties than one would classically expect.
Using simple combinatorial reasoning, two sorts of properties—say properties ‘M’ and ‘P’—could conceivably be related in only a few ways: they could be wholly distinct; all elements of ‘M’ could also be elements of ‘P’, or vice versa; there could be some kind of overlap; they could be identical; there could in fact be no ‘M’; or there could in fact be no ‘P’. In the philosophy of mind these correspond to the possibilities of dualism (parallelism), interactionism, dual-aspect theories, reductionism—both materialism and idealism—, and eliminativism (both of the materialist and the idealist sort, the latter of which nobody has, to my knowledge, ever defended).
Before the advent of quantum theory, I’m sure most philosophers would have considered that this list is exhaustive, and moreover, that it’s analytic that there couldn’t be any more options—that’s simply it, it’s gotta be one of those options, we only have to figure out which. But quantum mechanics hints at another option, typically called ‘complementarity’: two properties—say, wave-like and particle-like behavior—may be related in such a way that both are necessary for a complete explanation, yet both are mutually incompatible, and not reducible to one another.
Does this lead us anywhere? Well, who knows—but I think one important lesson is that new discoveries may not only introduce novel phenomena, but also, novel modes of understanding and explanation.
