  Understanding temporal processes and their correlations in time is of
paramount importance for the development of near-term technologies that operate
under realistic conditions. Capturing the complete multi-time statistics
defining a stochastic process lies at the heart of any proper treatment of
memory effects. In this thesis, using a novel framework for the
characterisation of quantum stochastic processes, we first solve the long
standing question of unambiguously describing the memory length of a quantum
processes. This is achieved by constructing a quantum Markov order condition
that naturally generalises its classical counterpart for the quantification of
finite-length memory effects. As measurements are inherently invasive in
quantum mechanics, one has no choice but to define Markov order with respect to
the interrogating instruments that are used to probe the process at hand:
different memory effects are exhibited depending on how one addresses the
system, in contrast to the standard classical setting. We then fully
characterise the structural constraints imposed on quantum processes with
finite Markov order, shedding light on a variety of memory effects that can
arise through various examples. Lastly, we introduce an instrument-specific
notion of memory strength that allows for a meaningful quantification of the
temporal correlations between the history and the future of a process for a
given choice of experimental intervention. These findings are directly relevant
to both characterising and exploiting memory effects that persist for a finite
duration. In particular, immediate applications range from developing efficient
compression and recovery schemes for the description of quantum processes with
memory to designing coherent control protocols that efficiently perform
information-theoretic tasks, amongst a plethora of others.
