The relationship between QCMA and QMA in the Turing and Communication model
Consider two complexity classes, say P and NP. To show that a proof that P=NP would need non-relativizing techniques, one has to describe an oracle relative to which the classes are not equal. For this the usual approach is to take an exponential separation between deterministic and non-deterministic query complexity and then use a standard argument to turn this into the oracle separation.

For algebrization one needs the stronger statement that even if the weaker class (here P) is given access to the polynomial extension of the oracle the classes remain different. Aaronson and Wigderson give some direct constructions for that, but the easiest argument is that a separation in communication complexity (e.g. an exponential separation between deterministic and non-deterministic communication complexity) can take the place of the query complexity separation. Compare Theorem 4.11 in the algebrization paper.

Of course many such separations are known (e.g. non-disjointness is easy for non-deterministic protocols but hard for deterministic protocols).
The separation that is left open in our paper is QCMA- vs. QMA-communication complexity (in the two-way model). Giving an exponential separation would immediately imply that any proof that QMA=QCMA (for Turing machines) would need non-algebrizing techniques (of course this inclusion might not be true).

Again, the reasoning here is that an algebrizing proof of QCMA=QMA would (by definiton) also work for the algebraic oracle model (for any oracle) and algorithms in this one could be simulated in the communication complexity model, contradicting a separation result there.