  Quantum mechanics has always proven emphatically as one of the main
cornerstones in all of the science since its inception. Initially, it has also
faced many skeptics from many scientific proponents of its complete description
of reality. However, John Bell once devised a theorem on quantitative grounds
to show how local realism is expressed in quantum mechanics. As Bell's
inequality claims the non-existence of local hidden variable theories, on the
same footing we have Leggett-Garg's Inequality (LGI), which sets a
quantum-classical limit for temporally correlated quantum systems. In the
context of particle physics specially in the field of neutrino and meson
oscillations, one can conveniently implement LGI to test the quantum
foundations at the probability level. Here, we discuss the significant LGI
violation characteristics in B- and K-mesons oscillations taking into account
their decoherence, CP violation and decay parameters. Also, we discuss and
comment on the behavior or effect of decoherence and decay widths playing out
from the perspective of LGI by taking their available values from various
experiments. This may help us to understand the underlying principles and
techniques of neutral meson open quantum systems.
