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

Macroscopic realism (MR) is the view that a system may possess definite properties at any time independent of past or future measurements and may be tested experimentally using the Leggett-Garg inequalities (LGIs). In this work we advance the study of LGIs in two ways using experiments carried out on a nuclear magnetic resonance spectrometer. First, we addresses the fact that the LGIs are only necessary conditions for MR but not sufficient ones. We implement a recently proposed test of necessary and sufficient conditions for MR which consists of a combination of the original four three-time LGIs augmented with a set of 12 two-time LGIs. We explore different regimes in which the two- and three-time LGIs may each be satisfied or violated. Second, we implement a recent proposal for a measurement protocol which determines the temporal correlation functions in an approximately noninvasive manner. It employs a measurement of the velocity of a dichotomic variable Q, continuous in time, from which a possible sign change of Q may be determined in a single measurement of an ancilla coupled to the velocity. This protocol involves a significantly different set of assumptions to the traditional ideal negative measurement protocol, and a comparison with the latter is carried out.