  Entanglement is a ubiquitous feature of low temperature systems and believed
to be highly relevant for the dynamics of condensed matter properties and
quantum computation even at higher temperatures. The experimental certification
of this paradigmatic quantum effect in macroscopic high temperature systems is
constrained by the limited access to the quantum state of the system. In this
paper we show how macroscopic observables beyond the energy of the system can
be exploited as proxy witnesses for entanglement detection. Using linear and
semi-definite relaxations we show that all previous approaches to this problem
can be outperformed by our proxies, i.e. entanglement can be certified at
higher temperatures without access to any local observable. For an efficient
computation of proxy witnesses one can resort to a generalized grand canonical
ensemble, enabling entanglement certification even in complex systems with
macroscopic particle numbers.
