  We formally extend the notion of Markov order to open quantum processes by
accounting for the instruments used to probe the system of interest at
different times. Our description recovers the classical Markov order property
in the appropriate limit: when the stochastic process is classical and the
instruments are non-invasive, \emph{i.e.}, restricted to orthogonal, projective
measurements. We then prove that there do not exist non-Markovian quantum
processes that have finite Markov order with respect to all possible
instruments; the same process exhibits distinct memory effects with respect to
different probing instruments. This naturally leads to a relaxed definition of
quantum Markov order with respect to specified sequences of instruments. The
memory effects captured by different choices of instruments vary dramatically,
providing a rich landscape for future exploration.
