  We introduce a new solution to Glauber multi-spin dynamics on random graphs.
The solution is based on the recently introduced Cavity Master Equation (CME),
a time-closure turning the in principle exact Dynamic Cavity Method into a
practical method of analysis and of fast simulation.
  Running CME once is of comparable computational complexity as one Monte Carlo
run on the same problem. We show that CME correctly models the ferromagnetic
$p$-spin Glauber dynamics from high temperatures down to and below the
spinoidal transition. We also show that CME allows a novel exploration of the
low-temperature spin-glass phase of the model.
