  We introduce a novel time-energy uncertainty relationship within the context
of restarts in monitored quantum dynamics. Initially, we investigate the
concept of ``first hitting time'' in quantum systems using an IBM quantum
computer and a three-site ring graph as our starting point. Previous studies
have established that the mean recurrence time, which represents the time taken
to return to the initial state, is quantized as an integer multiple of the
sampling time, displaying pointwise discontinuous transitions at resonances.
Our findings demonstrate that, the natural utilization of the restart mechanism
in laboratory experiments, driven by finite data collection time spans, leads
to a broadening effect on the transitions of the mean recurrence time. Our
newly proposed uncertainty relation captures the underlying essence of these
phenomena, by connecting the broadening of the mean hitting time near
resonances, to the intrinsic energies of the quantum system and to the
fluctuations of recurrence time. This work not only contributes to our
understanding of fundamental aspects related to quantum measurements and
dynamics, but also offers practical insights for the design of efficient
quantum algorithms with mid-circuit measurements.
