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Quantum mechanics has always proven emphatically as one of the main cornerstones in all of 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- meson oscillations, taking into account their decoherence, CP violation, and decay parameters. We have emphasized the fact that \textit{Tsirelson} bounds can be achieved under certain conditions. Our focal point is to show the signature of these bounds that appears only to specific alterations of decay and decoherence effects. 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.