  The Leggett-Garg inequality (LGI), based on the notions of realism and
noninvasive measurability, is applied in the context of a linear harmonic
oscillator. It is found that merely through observing at various instants which
region of the potential well, the oscillating quantum object is in, the LGI can
be violated without taking recourse to any ancillary quantum system.
Strikingly, this violation reveals an unexplored nonclassicality of the state
which is considered the most "classical-like" of all quantum states, namely the
Schr\"odinger coherent state. In the macrolimit, the extent to which such
nonclassicality persists for large values of mass and classical amplitudes of
oscillation is quantitatively investigated. It is found that while for any
given mass and oscillator frequency, a significant quantum violation of LGI can
be obtained by suitably choosing the initial peak momentum of the coherent
state wave packet, as the mass is sufficiently increased, actual observability
of this violation becomes increasingly difficult. A feasible experimental setup
for testing the predicted quantum mechanical violation of LGI is suggested
using a trapped nano-object of $\sim 10^6-10^9$ amu mass.

Quantum Physics