  We address two sets of long-standing open questions in probability theory,
from a computational complexity perspective: divisibility of stochastic maps,
and divisibility and decomposability of probability distributions. We prove
that finite divisibility of stochastic maps is an NP-complete problem, and
extend this result to nonnegative matrices, and completely-positive
trace-preserving maps, i.e. the quantum analogue of stochastic maps. We further
prove a complexity hierarchy for the divisibility and decomposability of
probability distributions, showing that finite distribution divisibility is in
P, but decomposability is NP-hard. For the former, we give an explicit
polynomial-time algorithm. All results on distributions extend to
weak-membership formulations, proving that the complexity of these problems is
robust to perturbations.
