  Markovian approximation is a widely-employed idea in descriptions of the
dynamics of open quantum systems (OQSs). Although it is usually claimed to be a
concept inspired by classical Markovianity, the term quantum Markovianity is
used inconsistently and often unrigorously in the literature. In this report we
compare the descriptions of classical stochastic processes and quantum
stochastic processes (as arising in OQSs), and show that there are inherent
differences that lead to the non-trivial problem of characterizing quantum
non-Markovianity. Rather than proposing a single definition of quantum
Markovianity, we study a host of Markov-related concepts in the quantum regime.
Some of these concepts have long been used in quantum theory, such as quantum
white noise, factorization approximation, divisibility, Lindblad master
equation, etc.. Others are first proposed in this report, including those we
call past-future independence, no (quantum) information backflow, and
composability. All of these concepts are defined under a unified framework,
which allows us to rigorously build hierarchy relations among them. With
various examples, we argue that the current most often used definitions of
quantum Markovianity in the literature do not fully capture the memoryless
property of OQSs. In fact, quantum non-Markovianity is highly
context-dependent. The results in this report, summarized as a hierarchy
figure, bring clarity to the nature of quantum non-Markovianity.
