  We use the quantum null energy condition in strongly coupled two-dimensional
field theories (QNEC2) as diagnostic tool to study a variety of phase
structures, including crossover, second and first order phase transitions. We
find a universal QNEC2 constraint for first order phase transitions with kinked
entanglement entropy and discuss in general the relation between the
QNEC2-inequality and monotonicity of the Casini-Huerta c-function. We then
focus on a specific example, the holographic dual of which is modelled by
three-dimensional Einstein gravity plus a massive scalar field with one free
parameter in the self-interaction potential. We study translation invariant
stationary states dual to domain walls and black branes. Depending on the value
of the free parameter we find crossover, second and first order phase
transitions between such states, and the c-function either flows to zero or to
a finite value in the infrared. Strikingly, evaluating QNEC2 for ground state
solutions allows to predict the existence of phase transitions at finite
temperature.
