  In recent years, the import of quantum information techniques in quantum
gravity opened new perspectives in the study of the microscopic structure of
spacetime. We contribute to such a program by establishing a precise
correspondence between the quantum information formalism of tensor networks
(TN), in the case of projected entangled-pair states (PEPS) generalised to a
second-quantized framework, and group field theory (GFT) states, and by showing
how, in this quantum gravity approach, discrete spatial manifolds arise as
entanglement patterns among quanta of space, having a dual representation in
terms of graphs and simplicial complexes. We devote special attention to the
implementation and consequences of the label independence of the
graphs/networks, corresponding to the indistinguishability of the space quanta
and representing a discrete counterpart of the diffeomorphism invariance of a
consistent quantum gravity formalism. We also outline a relational setting to
recover distinguishability of graph/network vertices at an effective and
physical level, in a partial semi-classical limit of the theory.
