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Macroscopic realism (MR) is the view that a system may possess definiteproperties at any time independent of past or future measurements, and may betested experimentally using the Leggett-Garg inequalities (LGIs). In this workwe advance the study of LGIs in two ways using experiments carried out on anuclear magnetic resonance spectrometer. Firstly, we addresses the fact thatthe LGIs are only necessary conditions for MR but not sufficient ones. Weimplement a recently-proposed test of necessary and sufficient conditions forMR which consists of a combination of the original four three-time LGIsaugmented with a set of twelve two-time LGIs. We explore different regimes inwhich the two- and three-time LGIs may each be satisfied or violated. Secondly,we implement a recent proposal for a measurement protocol which determines thetemporal correlation functions in an approximately non-invasive manner. Itemploys a measurement of the velocity of a dichotomic variable $Q$, continuousin time, from which a possible sign change of $Q$ may be determined in a singlemeasurement of an ancilla coupled to the velocity. This protocol involves asignificantly different set of assumptions to the traditional ideal negativemeasurement protocol and a comparison with the latter is carried out.

Exploration of an augmented set of Leggett-Garg inequalities using a noninvasive continuous-in-time velocity measurement