  We explore and contrast the capabilities of future colliders to probe the
nature of the electro-weak phase transition. We focus on the real singlet
scalar field extension of the Standard Model, representing the most minimal,
yet most elusive, framework that can enable a strong first-order electro-weak
phase transition. By taking into account the theoretical uncertainties and
employing the powerful complementarity between gauge and Higgs boson pair
channels in the searches for new scalar particles, we find that a 100 TeV
proton collider has the potential to confirm or falsify a strong first-order
transition. Our results hint towards this occurring relatively early in its
lifetime. Furthermore, by extrapolating down to 27 TeV, we find that a
lower-energy collider may also probe a large fraction of the parameter space,
if not all. Such early discoveries would allow for precise measurements of the
new phenomena to be obtained at future colliders and would pave the way to
definitively verify whether this is indeed the physical remnant of a scalar
field that catalyses a strong first-order transition.
