  We put the Density-of-States (DoS) approach to Monte-Carlo (MC) simulations
under a stress test by applying it to a physical problem with the worst
possible sign problem: the real time evolution of a non-integrable quantum spin
chain. Benchmarks against numerical exact diagonalisation and stochastic
reweighting are presented. Both MC methods, the DoS approach and reweighting,
allow for simulations of spin chains as long as $L=40$, far beyond exact
diagonalisability, though only for short evolution times $t\lesssim 1$. We
identify discontinuities of the density of states as one of the key problems in
the MC simulations and propose to calculate some of the dominant contributions
analytically, increasing the precision of our simulations by several orders of
magnitude. Even after these improvements the density of states is found highly
non-smooth and therefore the DoS approach cannot outperform reweighting. We
prove this implication theoretically and provide numerical evidence, concluding
that the DoS approach is not well suited for quantum real time simulations with
discrete degrees of freedom.
