Quantum mechanics as noncommutative probability theory?

From The Quantum Pontiff: Quantum mechanics as noncommutative probability theory?

As I pointed out in a comment there, there are quantum random walks, which are a generalization of classical random walks, but display quantum interference and "spread out" much faster than classical random walks, namely at a rate proportional to time, instead of proportional to the square root of time as in the case of classical random walks. The comments at the Quantum Pontiff, which address other aspects of quantum mechanics as noncommutative probability theory, could be worth a quick read if you're interested in this sort of thing. (I actually don't know enough quantum mechanics to comment intelligently.)

But ordinary ("commutative") probability theory is counterintuitive enough...