  We study the application of a new method for simulating nonlinear dynamics of
many-body spin systems using quantum measurement and feedback [Mu\~noz-Arias et
al., Phys. Rev. Lett. 124, 110503 (2020)] to a broad class of many-body models
known as $p$-spin Hamiltonians, which describe Ising-like models on a
completely connected graph with $p$-body interactions. The method simulates the
desired mean field dynamics in the thermodynamic limit by combining
nonprojective measurements of a component of the collective spin with a global
rotation conditioned on the measurement outcome. We apply this protocol to
simulate the dynamics of the $p$-spin Hamiltonians and demonstrate how
different aspects of criticality in the mean-field regime are readily
accessible with our protocol. We study applications including properties of
dynamical phase transitions and the emergence of spontaneous symmetry breaking
in the adiabatic dynamics of the collective spin for different values of the
parameter $p$. We also demonstrate how this method can be employed to study the
quantum-to-classical transition in the dynamics continuously as a function of
system size.
