  In stochastic modeling, the excess entropy -- the mutual information shared
between a processes past and future -- represents the fundamental lower bound
of the memory needed to simulate its dynamics. However, this bound cannot be
saturated by either classical machines or their enhanced quantum counterparts.
Simulating a process fundamentally requires us to store more information in the
present than than what is shared between past and future. Here we consider a
hypothetical generalization of hidden Markov models beyond classical and
quantum models -- n-machines -- that allow for negative quasi-probabilities. We
show that under the collision entropy measure of information, the minimal
memory of such models can equalize the excess entropy. Our results hint
negativity as a necessary resource for memory-advantaged stochastic simulation
-- mirroring similar interpretations in various other quantum information
tasks.
