  We show violations of Leggett-Garg inequalities to be possible for
single-mode cat-states evolving dynamically in the presence of a nonlinear
quantum interaction arising from, for instance, a Kerr medium. In order to
prove the results, we derive a generalised version of the Leggett-Garg
inequality involving different cat-states at different times. The violations
demonstrate failure of the premise of macro-realism as defined by Leggett and
Garg, provided extra assumptions associated with experimental tests are valid.
With the additional assumption of stationarity, violations of the Leggett-Garg
inequality are predicted for the multi-component cat-states observed in the
Bose-Einstein condensate and superconducting circuit experiments of Greiner et
al. (Nature 419, 51, (2002)) and Kirchmair et al. (Nature, 495, 205 (2013)).
The violations demonstrate a mesoscopic quantum coherence, by negating that the
system can be in a classical mixture of mesoscopically distinct coherent
states. Higher orders of nonlinearity are also studied and shown to give strong
violation of Leggett-Garg inequalities.
