  In contrast to the spatial Bell's inequalities, which probe entanglement
between spatially-separated systems, the Leggett-Garg inequalities test the
correlations of a single system measured at different times. Violation of a
genuine Leggett-Garg test implies either the absence of a realistic description
of the system or the impossibility of measuring the system without disturbing
it. Quantum mechanics violates the inequalities on both accounts and the
original motivation for these inequalities was as a test for quantum coherence
in macroscopic systems. The last few years has seen a number of experimental
tests and violations of these inequalities in a variety of microscopic systems
such as superconducting qubits, nuclear spins, and photons. In this article, we
provide an introduction to the Leggett-Garg inequalities and review these
latest experimental developments. We discuss important topics such as the
significance of the non-invasive measurability assumption, the clumsiness
loophole, and the role of weak measurements. Also covered are some recent
theoretical proposals for the application of Leggett-Garg inequalities in
quantum transport, quantum biology and nano-mechanical systems.
