If there were some quantum process involved in our neurons, the next question would be, when did it evolve? Is it in sponges? We actually have quite complete simulations of neural networks of some lower lifeforms. It must have then evolved sometime between the lifeforms we can simulate and ourselves. If that's the case, you'd expect some discontinuity in functionality vs volume. You don't see that anywhere.

And as stated in the article, it seems to be a solution in search of a problem.