Quantum violation of variants of LGIs upto algebraic maximum for qubit system

In 1985, Leggett and Garg formulated a class of inequalities for testing the compatibility between macrorealism and quantum mechanics. In this paper, we point out that based on the same assumptions of macrorealism that are used in the derivation of Leggett-Garg inequalities (LGIs) , there is a scope of formulating another class of inequalities different from standard LGIs. By considering the three-time measurement scenario in a dichotomic system, we first propose an interesting variant of standard LGIs and show that its quantum violation is larger than the standard LGI. By extending this formulation to $n$-time measurement scenario, we found that the quantum violations of variants of LGIs for a \emph{qubit} system increase with $n$, and for a sufficiently large $n$ algebraic maximum can be reached. Further, we compare the quantum violations of our formulated LGIs with the standard LGIs and no-signaling in time formulation of macrorealism.