# Inequivalent Leggett-Garg inequalities

## TL;DR

This paper demonstrates that Wigner form Leggett-Garg inequalities are fundamentally inequivalent and stronger than standard ones, providing insights into macrorealism and measurement disturbance in quantum systems.

## Contribution

The paper proves that Wigner form LGIs are inequivalent and stronger than standard LGIs, and explores their relation to no-signaling in time.

## Key findings

- Wigner form LGIs are inequivalent to standard LGIs.
- Wigner form LGIs are stronger than standard LGIs.
- Measurement disturbance quantifies inequality strength.

## Abstract

It remains an open question how realist view of macroscopic world emerges from quantum formalism. For testing the macrorealism in quantum domain, an interesting approach was put forward by Leggett and Garg in $1985$, by formulating a suitable inequality valid for any macrorealistic theory. Recently, by following the Wigner idea of local realist inequality, a probabilistic version of standard Leggett-Garg inequalities have also been proposed. While the Wigner form of local realist inequalities are equivalent to the two-party, two-measurements and two outcomes CHSH inequalities, in this paper we provide a generic proof to demonstrate that the Wigner form of Leggett-Garg inequalities are not only inequivalent to the standard ones but also stronger than the later. This is demonstrated by quantifying the amount of disturbance caused by a prior measurement to the subsequent measurements. In this connection, the relation between LGIs and another formulation of macrorealism known as no-signaling in time is examined.