The "canonical" coactions of compact quantum groups on the Cuntz algebras have been studied for some time with the focus on their fixed point algebras.
In this talk, I'll guess their crossed products and illustrate how my previous work on coactions of Hopf C*-algebras on C*-correspondences can be used to get their simplicity
when the compact quantum groups under consideration are R+ deformations of C(SU(2)) or C(SO(3)) in the sense of Banica, although this is probably well-known to some specialists.