  In numerical simulations of classical and quantum lattice systems, 2d corner
transfer matrices (CTMs) and 3d corner tensors (CTs) are a useful tool to
compute approximate contractions of infinite-size tensor networks. In this
paper we show how the numerical CTMs and CTs can be used, {\it additionally\/},
to extract universal information from their spectra. We provide examples of
this for classical and quantum systems, in 1d, 2d and 3d. Our results provide,
in particular, practical evidence for a wide variety of models of the
correspondence between $d$-dimensional quantum and $(d+1)$-dimensional
classical spin systems. We show also how corner properties can be used to
pinpoint quantum phase transitions, topological or not, without the need for
observables. Moreover, for a chiral topological PEPS we show by examples that
corner tensors can be used to extract the entanglement spectrum of half a
system, with the expected symmetries of the $SU(2)_k$ Wess-Zumino-Witten model
describing its gapless edge for $k=1,2$. We also review the theory behind the
quantum-classical correspondence for spin systems, and provide a new numerical
scheme for quantum state renormalization in 2d using CTs. Our results show that
bulk information of a lattice system is encoded holographically in
efficiently-computable properties of its corners.
