  Topological phases supporting non-abelian anyonic excitations have been
proposed as candidates for topological quantum computation. In this paper, we
study disordered non-abelian anyonic chains based on the quantum groups
$SU(2)_k$, a hierarchy that includes the $\nu=5/2$ FQH state and the proposed
$\nu=12/5$ Fibonacci state, among others. We find that for odd $k$ these
anyonic chains realize infinite randomness critical {\it phases} in the same
universality class as the $S_k$ permutation symmetric multi-critical points of
Damle and Huse (Phys. Rev. Lett. 89, 277203 (2002)). Indeed, we show that the
pertinent subspace of these anyonic chains actually sits inside the ${\mathbb
Z}_k \subset S_k$ symmetric sector of the Damle-Huse model, and this ${\mathbb
Z}_k$ symmetry stabilizes the phase.
