  Fractional entropy is a signature of nonlocal degrees of freedom, such as
Majorana zero modes or more exotic non-Abelian anyons. Although direct
experimental measurements remain challenging, Maxwell relations provide an
indirect route to the entropy through charge measurements. Here we consider
multichannel charge-Kondo systems, which are predicted to host exotic
quasiparticles due to a frustration of Kondo screening at low temperatures. In
the absence of experimental data for the charge occupation, we derive relations
connecting the latter to the conductance, for which experimental results have
recently been obtained. Our analysis indicates that Majorana and Fibonacci
anyon quasiparticles are well-developed in existing two- and three-channel
charge-Kondo devices, and that their characteristic $k_{\rm{B}}\log\sqrt{2}$
and $k_{\rm{B}}\log\frac{1+\sqrt{5}}{2}$ entropies are experimentally
measurable.
