  We introduce a framework to compute upper bounds for temporal correlations
achievable in open quantum system dynamics, obtained by repeated measurements
on the system. As these correlations arise by virtue of the environment acting
as a memory resource, such bounds are witnesses for the minimal dimension of an
effective environment compatible with the observed statistics. These witnesses
are derived from a hierarchy of semidefinite programs with guaranteed
asymptotic convergence. We compute non-trivial bounds for various sequences
involving a qubit system and a qubit environment, and compare the results to
the best known quantum strategies producing the same outcome sequences. Our
results provide a numerically tractable method to determine bounds on
multi-time probability distributions in open quantum system dynamics and allow
for the witnessing of effective environment dimensions through probing of the
system alone.
