  Some quantum information processing protocols necessitate quantum operations
that are invariant under complex conjugation. In this study, we analyze the
non-local resources necessary for implementing conjugation-symmetric
measurements on multipartite quantum networks. We derive conditions under which
a given multipartite conjugation can have locally implementable symmetric
measurements. In particular, a family of numbers called the ``magic-basis
spectrum'' comprehensively characterizes the local measurability of a given
2-qubit conjugation, as well as any other properties that are invariant under
local unitary transformations. We also explore the non-local resources required
for optimal measurements on known quantum sensor networks by using their
conjugation symmetries as a guide.
