We know that QRG with polynomial number of rounds is equal to EXP. And QRG with one round (i.e. two messages) is equal to PSPACE. But what about with a few messages?
Even the question is unknown with the classical class RG.
It is possible that QRG with a few rounds is already equal to EXP.
https://complexityzoo.net/Complexity_Zoo:Q#qrg