  As is well known, the macroscopic realism and the noninvasive measurability
together lead to Leggett-Garg inequalities violated by quantum mechanics. We
consider tests of the Leggett-Garg type with use of the $q$-entropies. For all
$q\geq1$, quantum mechanics predicts violations of an entire family of
$q$-entropic inequalities of the Leggett-Garg type. Violations are exemplified
with a quantum spin-$s$ system. In general, entropic Leggett-Garg inequalities
give only necessary conditions that some probabilistic model is compatible with
the macrorealism in the broader sense. The presented $q$-entropic inequalities
allow to widen a class of situations, in which an incompatibility with the
macrorealism can be tested. In the considered example, both the strength and
range of violations are somehow improved by varying $q$. We also examine
$q$-entropic inequalities of the Leggett-Garg type in the case of detection
inefficiencies, when the no-click event may occur in each observation. With the
use of the $q$-entropic inequalities, the required amount of efficiency may be
reduced.
