  Many composite Higgs models predict the existence of vector-like quarks with
masses outside the reach of the LHC, e.g. $m_Q \gtrsim 2$ TeV, in particular if
these models contain a dark matter candidate. In such models the mass of the
new resonances is bounded from above to satisfy the constraint from the
observed relic density. We therefore develop new strategies to search for
vector-like quarks at a future $100$ TeV collider and evaluate what masses and
interactions can be probed. We find that masses as large as $\sim 6.4$ ($\sim
9$) TeV can be tested if the fermionic resonances decay into Standard Model
(dark matter) particles. We also discuss the complementarity of dark matter
searches, showing that most of the parameter space can be closed. On balance,
this study motivates further the consideration of a higher-energy hadron
collider for a next generation of facilities.
