  Leggett and Garg formulated macrorealist models encoding our intuition on
classical systems, i.e., physical quantities have a definite value that can be
measured with minimal disturbance, and with the goal of testing macroscopic
quantum coherence effects. The associated inequalities, involving the
statistics of sequential measurements on the system, are violated by quantum
mechanical predictions and experimental observations. Such tests, however, are
subject to loopholes: a classical explanation can be recovered assuming
specific models of measurement disturbance. We review recent theoretical and
experimental progress in characterizing macrorealist and quantum temporal
correlations, and in closing loopholes associated with Leggett-Garg tests.
Finally, we review recent definitions of nonclassical temporal correlations,
which go beyond macrorealist models by relaxing the assumption on the
measurement disturbance, and their applications in sequential quantum
information processing.
