  The $Q$-prime curvature is a local pseudo-Einstein invariant on CR manifolds
defined by Case and Yang, and Hirachi. Its integral, the total $Q$-prime
curvature, gives a non-trivial global CR invariant. On the other hand, Marugame
has constructed a family of global CR invariants via renormalized
characteristic forms, which contains the total $Q$-prime curvature. In this
paper, we introduce a generalization of the $Q$-prime curvature for each
renormalized characteristic form, and show that its integral coincides with
Marugame's CR invariant. We also study generalizations of the critical CR GJMS
operator and the $P$-prime operator, which are related to the transformation
laws of our new curvatures under conformal change.
