  We show that in some cases the coherent state can have a larger violation of
the Leggett-Garg inequality (LGI) than the cat state by numerical calculations.
To achieve this result, we consider the LGI of the cavity mode weakly coupled
to a zero-temperature environment as a practical instance of the physical
system. We assume that the bosonic mode undergoes dissipation because of an
interaction with the environment but is not affected by dephasing. Solving the
master equation exactly, we derive an explicit form of the violation of the
inequality for both systems prepared initially in the coherent state
$|\alpha\rangle$ and the cat state $(|\alpha\rangle+|-\alpha\rangle)$. For the
evaluation of the inequality, we choose the displaced parity operators
characterized by a complex number $\beta$. We look for the optimum parameter
$\beta$ that lets the upper bound of the inequality be maximum numerically.
Contrary to our expectations, the coherent state occasionally exhibits quantum
quality more strongly than the cat state for the upper bound of the violation
of the LGI in a specific range of three equally spaced measurement times
(spacing $\tau$). Moreover, as we let $\tau$ approach zero, the optimized
parameter $\beta$ diverges and the LGI reveals intense singularity.
