  In this work we study temporal quantum correlations, quantified by
Leggett-Garg (LG) and LG-type inequalities, in the $B$ and $K$ meson systems.
We use the tools of open quantum systems to incorporate the effect of
decoherence which is quantified by a single phenomenological parameter. The
effect of $CP$ violation is also included in our analysis. We find that the LG
inequality is violated for both $B$ and $K$ meson systems, the violation being
most prominent in the case of $K$ mesons and least for $B_s$ system. Since the
systems with no coherence do not violate LGI, incorporating
\textit{decoherence} is expected to decrease the extent of violation of LGI and
is clearly brought out in our results. We show that the expression for the LG
functions depends upon an additional term, apart from the experimentally
measurable meson transition probabilities. This term vanishes in the limit of
zero decoherence. On the other hand, the LG-type parameter can be directly
expressed in terms of transition probabilities, making it a more appropriate
observable for studying temporal quantum correlations in neutral meson systems.
