Rare top quark decays at a 100 TeV proton-proton collider: $t\to bWZ$ and $t\to hc$

We investigate extremely rare top quark decays at a future proton-proton collider with centre-of-mass energy of 100 TeV. We focus on two decay modes: radiative decay with a $Z$ boson, $t\to bWZ$, and flavour-changing neutral decay with a Higgs boson, $t\to hc$, the former being kinematically suppressed with a branching ratio of $\mathcal{O}({10}^{-6})$, and the latter highly loop-suppresed, with a branching ratio of $\mathcal{O}({10}^{-15})$. We find that $t\to bWZ$ will be very challenging to observe in top quark pair production, even within well-motivated beyond-the-Standard Model scenarios. For the mode $t\to hc$ we find a stronger sensitivity than that obtained by any future LHC measurement by at least one order of magnitude.