  We use extensive numerical simulations based on matrix product state methods
to study the quantum dynamics of spin chains with strong on-site disorder and
power-law decaying ($1/r^\alpha$) interactions. We focus on two spin-$1/2$
Hamiltonians featuring power-law interactions: Heisenberg and XY and
characterize their corresponding long-time dynamics using three distinct
diagnostics: decay of a staggered magnetization pattern $I(t)$, growth of
entanglement entropy $S(t)$, and growth of quantum Fisher information $F_Q(t).$
For sufficiently rapidly decaying interactions $\alpha>\alpha_c$ we find a
many-body localized phase, in which $I(t)$ saturates to a non-zero value,
entanglement entropy grows as $S(t)\propto t^{1/\alpha}$, and Fisher
information grows logarithmically. Importantly, entanglement entropy and Fisher
information do not scale the same way (unlike short range interacting models).
The critical power $\alpha_c$ is smaller for the XY model than for the
Heisenberg model.
