There is no role insofar as anyone knows. On the other hand, there are reasons to believe that quantum-mechanical coherence may be relevant to the functioning of the human brain. Professor Roger Penrose is one of the main proponents of this idea, discussed in particular in the second of the following two books:
1. The Emperor's New Mind: Concerning Computers, Minds and The Laws of Physics (1989).
2. Shadows of the Mind: A Search for the Missing Science of Consciousness (1994).
In the attached short video, Professor Penrose highlights some of his ideas on the subject matter.