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There have been some poorly received hypotheses to this effect, most notably Roger Penrose's Orch-OR: https://en.wikipedia.org/wiki/Orchestrated_objective_reducti...

That said, most attempts at quantifying whether or not distinctly quantum mechanical processes in the brain related to things like microtubules and NMDA receptors are significant to cognition (i.e. is the brain a quantum computer?) have generally concluded the answer is no:

See:

https://arxiv.org/abs/quant-ph/9907009

https://onlinelibrary.wiley.com/doi/pdf/10.1207/s15516709cog...