  How a closed interacting quantum many-body system relaxes and dephases as a
function of time is a fundamental question in thermodynamic and statistical
physics. In this work, we analyse and observe the persistent temporal
fluctuations after a quantum quench of a tunable long-range interacting
transverse-field Ising Hamiltonian realized with a trapped-ion quantum
simulator. We measure the temporal fluctuations in the average magnetization of
a finite-size system of spin-$1/2$ particles. We experiment in a regime where
the properties of the system are closely related to the integrable Hamiltonian
with global spin-spin coupling, which enables analytical predictions even for
the long-time non-integrable dynamics. The analytical expression for the
temporal fluctuations predicts the exponential suppression of temporal
fluctuations with increasing system size. Our measurement data is consistent
with our theory predicting the regime of many-body dephasing.
