  In this note, we analyze joint probability distributions that arise from
outcomes of sequences of quantum measurements performed on sets of quantum
states. First, we identify some properties of these distributions that need to
be fulfilled to get a classical behavior. Secondly, we prove that a joint
distribution exists iff measurement operators "on-state" permute (permutability
is the commutativity of more than two operators). By "on-state" we mean
properties of operators that hold only on a subset of states in the Hilbert
space. Then, we disprove a conjecture proposed by
  Carstens, Ebrahimi, Tabia, and Unruh (eprint 2018), which states that the
property of partial on-state permutation implies full on-state permutation. We
disprove this conjecture with a counterexample where pairwise "on-state"
commutativity does not imply on-state permutability, unlike in the case of
commutativity for all states in the Hilbert space.
  Finally, we explore the new concept of on-state commutativity by showing a
simple proof that if two projections almost on-state commute, then there is a
commuting pair of operators that are on-state close to the originals. This
result was originally proven by Hastings (Communications in Mathematical
Physics, 2019) for general operators.
