  We study the spectrum of multiple non-Abelian anyons in a harmonic trap. The
system is described by Chern-Simons theory, coupled to either bosonic or
fermionic non-relativistic matter, and has an SO(2,1) conformal invariance. We
describe a number of special properties of the spectrum, focussing on a class
of protected states whose energies are dictated by their angular momentum. We
show that the angular momentum of a bound state of non-Abelian anyons is
determined by the quadratic Casimirs of their constituents.
