(Assuming Mind=Brain, i.e. the entire mind is just the physical brain and no "soul" is involved. Also, Neurons aren't really all that small, they're quite macroscopic -- though the processes in the neurons like chemical interactions need quantum mechanics for their description)

In Newtonian Mechanics, it is sufficient to know the positions and velocities (i.e. derivaties of position) of particles to determine future states. So, the world is Markov given this informatio.

In Schrodinger's equation, you again only need to know \Psi and it's time derivative to know all future states. I think the quantum properties of the brain are adequately described just with Schodinger's equation. You do need to include nuclear forces etc in a description of the brain. You may need quantum electrodynamics, but I think Schrodinger's equation is sufficient.

My physics education stopped before I got here, but Dirac's equation which may be necessary to model the brain seems to require the second time-derivative of the wavefunction -- so you may need the second order time-derivatives to make the model Markov. Can someone who knows a bit more quantum physics chime in here?

EDIT: Reading the wiki article more carefully, it seems Dirac's equation is also first order

In the model there's the distribution p, which determines how the world is changing. In the chess example this would include: a) how the agent's action changes the state of the game + b) some distribution we assume (but which we may or may not actually know) about the opponent's action and the resulting state of the game. In a physics example, p should include the relevant laws of physics, together with constants which tell the rate (and manner) in which the world is changing. Any changing parameters should be part of the state.

It seems that you're saying that it may be difficult to know what p is. Then you are very much correct. You probably couldn't infer the laws of physics from the current wave function of the universe, or the rules of chess from the current state of the game. But at this point we're only assuming that such laws exist, not that we know how to learn them.

p and q are probability distributions, which is where we allow for randomness in the process. But note that randomness becomes a tricky concept if you go deep enough into physics.

As for the "quantum mind" theory, as far as I can tell it's fringe science at best. Personally, I'm very skeptical. Regardless, such a model can still have the Markov property, if you include the wave function in your state.