Complexity analysis of separability in the multipartite case

Question

It's well known that determining whether a bipartite mixed state is separable or entangled is a $\mathsf{NP}$-hard problem under some accuracy estimates (cf. this TCS SE discussion). Now I'm curious whether there exist detailed treatises on the complexity analysis of various possible cases of complete¹ and partial separability of multipartite (pure and mixed) states.

Could someone provide some insight into this? Say, is there any specific case of partial separability, determining which would be a problem in $\mathsf{P}$ rather than in $\mathsf{NP}$?

^{1: For complete separability, I believe the same arguments apply as for the bipartite case, as mentioned by Norbert Schuch. However, it'd be interesting to look at partial separability too.}