  It has recently been reported [\textit{PNAS} \textbf{114}, 2303 (2017)] that,
under an operational definition of time, quantum clocks would get entangled
through gravitational effects. Here we study an alternative scenario: the
clocks have different masses and energy gaps, which would produce time
difference via gravitational interaction. The proposal of quantum clock
synchronization for the gravity-induced time difference is discussed. We
illustrate how the stability of measurement probability in the quantum clock
synchronization proposal is influenced by the gravitational interaction induced
by the clock themselves. It is found that the precision of clock
synchronization depends on the energy gaps of the clocks and the improvement of
precision in quantum metrology is in fact an indicator of entanglement
generation. We also present the quantum enhanced estimation of time difference
and find that the quantum Fisher information is very sensitive to the distance
between the clocks.
