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We provide bounds on temporal fluctuations around the infinite-time averageof out-of-time-ordered and time-ordered correlators of many-body quantumsystems without energy gap degeneracies. For physical initial states, ourbounds predict the exponential decay of the temporal fluctuations as a functionof the system size. We numerically verify this prediction for chaotic andinteracting integrable spin-1/2 chains, which satisfy the assumption of ourbounds. On the other hand, we show analytically and numerically that for the XXmodel, which is a noninteracting system with gap degeneracies, the temporalfluctuations decay polynomially with system size for operators that are localin the fermion representation and decrease exponentially in the system size fornon-local operators. Our results demonstrate that the decay of the temporalfluctuations of correlators cannot be used as a reliable metric of chaos orlack thereof.

Temporal fluctuations of correlators in integrable and chaotic quantum systems