  In classical physics, memoryless dynamics and Markovian statistics are one
and the same. This is not true for quantum dynamics, first and foremost because
quantum measurements are invasive. Going beyond measurement invasiveness, here
we derive a novel distinction between classical and quantum processes, namely
the possibility of hidden quantum memory. While Markovian statistics of
classical processes can always be reproduced by a memoryless dynamical model,
our main result shows that this is not true in quantum mechanics: We first
provide an example of quantum non-Markovianity whose manifestation depends on
whether or not a previous measurement is performed -- an impossible phenomenon
for memoryless dynamics; we then strengthen this result by demonstrating
statistics that are Markovian independent of how they are probed, but are
nonetheless still incompatible with memoryless quantum dynamics. Thus, we
establish the existence of Markovian statistics gathered by probing a quantum
process that nevertheless fundamentally require memory for their creation.
