CANCELED - Given a tensor network description of a many-body quantum state, can we calculate quantum state amplitudes? For translation-invariant (TI) systems, this is a question of the computational resources necessary to contract a TI tensor network. In d spatial dimensions, we can address this question by studying non-unitary ‘dynamics’ generated by the action of a fixed transfer matrix acting on a (d-1)-dimensional system. If this (d-1)-dimensional system becomes highly entangled, the classical memory required to calculate amplitudes can grow exponentially with system size, i.e. there is a computational barrier between different descriptions of our state. Such a barrier would suggest the possibility of an entanglement transition in the dynamics when tensor network parameters are varied; unfortunately, while entanglement transitions in random tensor networks can be understood through a mapping to statistical mechanics, this formalism cannot be leveraged in TI systems. In this talk I will present evidence for entanglement transitions in TI systems and show how these transitions are encoded in the spectrum of the transfer matrix. To describe the ‘entangling phase’ I will present exact results for spectral properties of non-unitary random matrices and show that, more generally, the stability of this phase is a consequence of level attraction along the radial direction in the complex plane.