  Understanding the demarcation line between classical and quantum is an
important issue in modern physics. The development of such an understanding
requires a clear picture of the various concurrent notions of `classicality' in
quantum theory presently in use. Here, we focus on the relationship between
Kolmogorov consistency of measurement statistics -- the foundational footing of
classical stochastic processes in standard probability theory -- and the
commutativity (or absence thereof) of measurement operators -- a concept at the
core of quantum theory. Kolmogorov consistency implies that the statistics of
sequential measurements on a (possibly quantum) system could be explained
entirely by means of a classical stochastic process, thereby providing an
operational notion of classicality. On the other hand, commutativity of
measurement operators is a structural property that holds in classical physics
and its breakdown is the origin of the uncertainty principle, a fundamentally
quantum phenomenon. Here, we formalise the connection between these two a
priori independent notions of classicality, demonstrate that they are distinct
in general and detail their implications for memoryless multi-time quantum
processes.
