We don't really understand consciousness, but many people have tried. A compelling argument pushed by Douglas Hofstadter is that consciousness is a product of certain sufficiently complex "strange loops", self-reflexive processes. Put simply, the mind observes the body, but it also observes itself observing the body, and it observes itself observing itself observing the body, and so on. On some level, this endlessly reflexive process of self-observation is consciousness.
Part of this self-reflexive process is the simulation of simplified models of the world, including ourselves and other people. This allows us to anticipate the effects of possible actions, which helps us choose the actions that we ultimately take. (Much or even most of our daily decision making doesn't use this complicated process and simply uses subconscious pattern matching. But we're focusing on the conscious part of the mind here.) Running multiple simulations to choose from a set of possible actions requires starting the simulation with the same initial state multiple times. In other words, it requires copying the initial state.
If consciousness were a quantum phenomenon, this would imply copying some quantum states. However, it is physically impossible to copy a quantum state due to the no-cloning theorem, just like it is impossible to change the amount of energy in the universe.
This immediately gives us a very nice explanation for why we are unable to perceive quantum states. Nothing can copy quantum states, and so a conscious mind that perceives quantum states cannot develop, since perception and memory of states of the world certainly requires making copies of those states (at least partial ones).
There is some wiggle-room in this argument. Even a mind that only perceives, remembers, and computes on internal representations of classical states might still use quantum physical processes for this computation. And at a certain level, it is actually obviously true that our brains do this: our brains are full of chemistry, and chemistry is full of quantum physics. The point is that this is just an irrelevant implementation detail. Those quantum processes could just as well be simulated by something classical.
And keep in mind that if we believe in the importance of "strange loops", then the mind itself is one of the things that the mind perceives. The mind cannot perceive quantum states, and so any kind of quantum-ness in the operation of the mind must be purely incidental. It cannot be a part of the "strange loop", and so it must ultimately be of limited importance.
There is yet more wiggle-room. We know that some computational tasks can be implemented much more efficiently on a quantum computer than on a classical computer - at least in theory. Nobody has ever built a true quantum computer that is big enough to demonstrate a speed-up over classical computers and
I doubt it. It looks increasingly like even in theory, quantum computers can only speed up a few very specialized mathematical problems, and that's not what our minds are particularly good at.
It's always possible that this whole argument will turn out to be wrong, once we learn more about consciousness in the future (and wouldn't that be interesting!), but for now, the bottom line for me is: Strange information processing loops most likely play an important role in consciousness. Such loops cannot involve quantum states due to the no-cloning theorem. Hence, consciousness is not a quantum phenomenon.