  The viability of non-stoquastic catalyst Hamiltonians to deliver consistent
quantum speedups in quantum adiabatic optimization remains an open question.
The infinite-range ferromagnetic $p$-spin model is a rare example exhibiting an
exponential advantage for non-stoquastic catalysts over its stoquastic
counterpart. We revisit this model and note how the incremental changes in the
ground state wavefunction give an indication of how the non-stoquastic catalyst
provides an advantage. We then construct two new examples that exhibit an
advantage for non-stoquastic catalysts over stoquastic catalysts. The first is
another infinite range model that is only 2-local but also exhibits an
exponential advantage, and the second is a geometrically local Ising example
that exhibits a growing advantage up to the maximum system size we study.
