  We investigate the nonequilbrium dynamics of entanglement entropy and
out-of-time-order correlator (OTOC) of noninteracting fermions at half-filling
starting from a product state to distinguish the delocalized, multifractal (in
the limit of nearest neighbor hopping), localized and mixed phases hosted by
the quasiperiodic Aubry-Andr\'e-Harper (AAH) model in the presence of
long-range hopping. For sufficiently long-range hopping strength a secondary
logarithmic behavior in the entanglement entropy is found in the mixed phases
whereas the primary behavior is a power-law the exponent of which is different
in different phases. The saturation value of entanglement entropy in the
delocalized, multifractal and mixed phases depends linearly on system size
whereas in the localized phase (in the short-range regime) it is independent of
system size. The early-time growth of OTOC shows very different power-law
behaviors in the presence of nearest neighbor hopping and long-range hopping.
The late time decay of OTOC leads to noticeably different power-law exponents
in different phases. The spatial profile of OTOC and its system-size dependence
also provide distinct features to distinguish phases. In the mixed phases the
spatial profile of OTOC shows two different dependences on space for small and
large distances respectively. Interestingly the spatial profile contains large
fluctuations at the special locations related to the quasiperiodicity parameter
in the presence of multifractal states.
