# Comparing Conditions for Macrorealism: Leggett-Garg Inequalities vs No-Signaling in Time

## TL;DR

This paper compares Leggett-Garg inequalities and no-signaling in time conditions as tests for macrorealism, showing their differences in necessity, sufficiency, and the strength of the classicality conditions they impose.

## Contribution

It demonstrates that augmented LG inequalities are necessary and sufficient for macrorealism, while NSIT conditions serve as stronger coherence witnesses, highlighting different notions of macrorealism.

## Key findings

- Augmented LG inequalities are necessary and sufficient for macrorealism.
- NSIT conditions are stronger coherence witnesses requiring zero interference.
- LG tests can be satisfied with moderate interference effects.

## Abstract

We consider two different types of conditions which were proposed to test macrorealism in the context of a system described by a single dichotomic variable Q. This is the view that a macroscopic system evolving in time possesses definite properties which can be determined without disturbing the future or past state. The Leggett-Garg (LG) inequalities, the most commonly-studied test, are only necessary conditions for macrorealism, but building on earlier work it is shown that when the four three-time LG inequalities are augmented with a certain set of two-time inequalities also of the LG form, Fine's theorem applies and these augmented conditions are then both necessary and sufficient. A comparison is carried out with a very different set of necessary and sufficient conditions for macrorealism, namely the no-signaling in time (NSIT) conditions proposed by Brukner, Clemente, Kofler and others, which ensure that all probabilities for Q at one and two times are independent of whether earlier or intermediate measurements are made in a given run, and do not require (but imply) the LG inequalities. We argue that tests based on the LG inequalities have the form of very weak classicality conditions and can be satisfied in the face of moderate interference effects, but those based on NSIT conditions have the form of much stronger coherence witness conditions, satisfied only for zero interference. The two tests differ in their implementation of non-invasive measurability so are testing different notions of macrorealism: the augmented LG tests are indirect, entailing a combination of the results of different experiments with only compatible quantities measured in each experimental run, in close analogy with Bell tests, and are primarily tests for macrorealism per se; by contrast the NSIT tests entail sequential measurements of incompatible quantities and are primarily tests for non-invasiveness