  The Leggett-Garg inequalities impose restrictions on the values taken by some
combinations of the two-time correlation functions of observables in order to
be explainable by a noninvasive-realist classical model. While in the unitary
dynamics, it is straightforward to compute these correlation functions, open
system effects bring in subtleties. Specifically, for non-Markovian dynamics,
which involves setting up of system-bath correlations, the Leggett-Garg
measurements disrupt these correlations, making a full system-bath Hamiltonian
approach natural. However, here we point out that the problem can also be dealt
with from a reduced dynamics perspective. The key point is that the noise
superoperator acting on the system must be suitably updated after measurement
interventions. Also considered is the effect of Markovian versus non-Markovian
behavior as well as classically non-Markovian processes on the violation of
Leggett-Garg inequalities.
