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 to derive Leggett-Garg inequalities (LGIs), there is a scope of formulating another class of inequalities different from the standard LGIs. By considering the three-time measurement scenario in a dichotomic system, we first propose an interesting variant of the standard LGIs and show that its quantum violation is larger than the standard LGIs. By extending this formulation to the $n$-time measurement scenario, we found that the quantum violations of variants of the LGIs for a qubit system increase with $n$, and, for a sufficiently large $n$, an algebraic maximum can be reached. We then compare the violations of the standard and variants of the LGIs in an unsharp measurement scenario and show that, for any arbitrary $n$, the violation of the latter is more robust to unsharpness than the former. Furthermore, we examine the relation between the quantum violations of the variants of the LGIs and an another formulation macrorealism, known as no-signaling in time conditions.

• Received 21 July 2018
• Revised 20 October 2018

DOI:https://doi.org/10.1103/PhysRevA.98.062115