  We study the phenomenology of a strongly-interacting top quark at future
hadron and lepton colliders, showing that the characteristic four-top contact
operators give rise to the most significant effects. We demonstrate the
extraordinary potential of a 100 TeV proton-proton collider to directly test
such non-standard interactions in four-top production, a process that we
thoroughly analyze in the same-sign dilepton and trilepton channels, and
explore in the fully hadronic channel. Furthermore, high-energy
electron-positron colliders, such as CLIC or the ILC, are shown to exhibit an
indirect yet remarkable sensitivity to four-top operators, since these
constitute, via renormalization group evolution, the leading new-physics
deformations in top-quark pair production. We investigate the impact of our
results on the parameter space of composite Higgs models with a
strongly-coupled (right-handed) top quark, finding that four-top probes provide
the best sensitivity on the compositeness scale at the future energy frontier.
In addition, we investigate mild yet persisting LHC excesses in multilepton
plus jets final states, showing that they can be consistently described in the
effective field theory of such a new-physics scenario.
