  In this paper we first compute the out-of-time-order correlators (OTOC) for
both a phenomenological model and a random-field XXZ model in the many-body
localized phase. We show that the OTOC decreases in power law in a many-body
localized system at the scrambling time. We also find that the OTOC can also be
used to distinguish a many-body localized phase from an Anderson localized
phase, while a normal correlator cannot. Furthermore, we prove an exact theorem
that relates the growth of the second R\'enyi entropy in the quench dynamics to
the decay of the OTOC in equilibrium. This theorem works for a generic quantum
system. We discuss various implications of this theorem.
