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Cool question! I asked the exact same question to a Professor of Quantum Physics when we were at a nanotechnology conference together. Obviously chemistry is based on quantum physics, etc etc. but really the spirit of what you’re asking is if neuron action potentials are in some way gated by a superposition of states effect which causes the human brain to be more advanced than if it was just action potentials described by classical E&M. It turns out that quantum effects are not necessary as far as we can tell, but that some scientists are looking for an as-of-yet-undiscovered quantum computer type processor in neurons.
My guess is that the human brain can operate within two orders of magnitude of the Landauer limit (compared to 6 for current silicon) and that’s why certain tasks like self driving cars are struggling so much to meet human performance without exorbitant energy expenditures. Basically our brains are remarkably energy efficient processors!
There's nothing here!