As discrete implementations of the anti de-Sitter/conformal field theory (AdS/CFT) correspondence, hyperinvariant tensor networks have created bridges between the fields of quantum information theory and quantum gravity. Adding SU(2) symmetry to the tensor network allows a direct connection to spin network states, a basis of the kinematic Hilbert space of loop quantum gravity (LQG). I discuss existence and limitations for hyperinvariant tensor networks incorporating a local SU(2) symmetry. I show that important aspects of the AdS/CFT correspondence are realized in certain quantum states of the gravitational field in LQG, thus justifying, from first principles, a class of models introduced by [F. Pastawski et al., JHEP 06, 149 (2015)]. I provide examples of hyperinvariant tensor networks, but also prove constraints on their existence in the form of no-go theorems that exclude absolutely maximally entangled states as well as general holographic codes from local SU(2)-invariance. I finally discuss applications of this new connection and existing examples in the form of calculations of surface areas and geodesic lengths as expectation values of the LQG area and length operators.