  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
