Unfortunately the physics underlying brain function are chaotic systems, meaning infinite (or “maximum”) precision is required to ensure two systems evolve to the same later states.
That level of precision cannot be achieved in measuring the state, without altering the state into something unknown after the moment of measurement.
Nothing quantum is necessary for this inability to determine state. Consider the problem of trying to map out where the eight ball is on a pool table, but you can’t see the eight ball. All you can do is throw other balls at it and observe how their velocities change. Now imagine you can’t see those balls either, because the sensing mechanism you’re using is composed of balls of equal or greater size.
Unsolvable problem. Like a box trying to contain itself.