  The sign problem is a key challenge in computational physics, encapsulating
our inability to properly understand many important quantum many-body phenomena
in physics, chemistry and the material sciences. Despite its centrality, the
circumstances under which the problem arises or can be resolved as well as its
interplay with the related notion of `non-stoquasticity' are often not very
well understood. In this study, we make an attempt to elucidate the
circumstances under which the sign problem emerges and to clear up some of the
confusion surrounding this intricate computational phenomenon. To that aim, we
make use of the recently introduced off-diagonal series expansion quantum Monte
Carlo scheme with which we analyze in detail a number of examples that capture
the essence of our results.
