Toric degenerations of schemes are a way to replace geometric or algebraic questions with questions about polyhedral geometry. In this talk we discuss how the combinatorics of objects from mathematical physics, the conformal blocks, can be used to construct flat degenerations of the Cox ring of the moduli of quasi-parabolic principal bundles on an n-marked curve of genus g. We will discuss when these degenerations are toric, and how the resulting combinatorial pictures can be used to prove structural theorems about this ring.