  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. Firstly, 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 twelve two-time LGIs. We explore different regimes in
which the two- and three-time LGIs may each be satisfied or violated. Secondly,
we implement a recent proposal for a measurement protocol which determines the
temporal correlation functions in an approximately non-invasive 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.
