  Non-Markovian quantum processes exhibit different memory effects when
measured in different ways; an unambiguous characterization of memory length
requires accounting for the sequence of instruments applied to probe the system
dynamics. This instrument-specific notion of quantum Markov order displays
stark differences to its classical counterpart. Here, we explore the structure
of quantum stochastic processes with finite length memory in detail. We begin
by examining a generalized collision model with memory, before framing this
instance within the general theory. We detail the constraints that are placed
on the underlying system-environment dynamics for a process to exhibit finite
Markov order with respect to natural classes of probing instruments, including
deterministic (unitary) operations and sequences of generalized quantum
measurements with informationally-complete preparations. Lastly, we show how
processes with vanishing quantum conditional mutual information form a special
case of the theory. Throughout, we provide a number of representative,
pedagogical examples to display the salient features of memory effects in
quantum processes.
