  Quantum entanglement is a key ingredient for quantum information processing
with capabilities beyond that of classical computation. We study the generation
and role of entanglement in the dynamics of spin-1/2 models, both for the
design of quantum gates for general-purpose quantum computation and for quantum
simulation of interacting spin models. We introduce the neutral atom
M{\o}lmer-S{\o}rensen gate, involving rapid adiabatic Rydberg dressing
interleaved in a spin-echo sequence. We show its robustness to quasi-static
experimental imperfections and favorable scaling with the time-energy scales of
Rydberg-mediated entanglement generation. In quantum simulation, we consider
critical behavior in quench dynamics of transverse field Ising models. Using
matrix product states to calculate the dynamics, we find that order parameters,
critical point, and critical exponents can be estimated using modest bond
dimensions. Considering the role of chaos and equilibration in quenches, we
find that local observables are well approximated either due to low global
entanglement or the proximity of local marginals to the maximally mixed state.
These findings highlight the challenge of identifying relevant quantum
phenomena that remain inaccessible to classical descriptions. Understanding the
regimes where classical descriptions fail but remain accessible to pre-fault
tolerant quantum hardware will help inform the design of future quantum
information processors
