  The quantum null energy condition (QNEC) is the only known consistent local
energy condition in quantum theories. Contrary to the classical energy
condition which are known to be violated in QFT, QNEC is a consequence of the
quantum focussing conjecture and has been proven for several special cases and
in general for QFTs in three or more spacetime dimensions. QNEC involves an
intrinsically quantum property of the theory under consideration, the
entanglement entropy (EE). While EE is notoriously hard to calculate in QFT,
the holographic principle provides a simple geometric description. In general
the holographic principle relates a gauge theory without gravity to a theory of
quantum gravity in one dimension higher. Holography provides a way to learn
about strongly coupled field theories as well as quantum gravity and
investigating QNEC in this context will undoubtedly lead to new insights. In
this thesis the focus is put on $2$- and $4$-dimensional field theories, where
we study systems of increasing complexity with numerical and analytical
methods. In vacuum, thermal states, globally quenched states and a toy model
for heavy ion collisions we find that QNEC is always satisfied and sometimes
saturated, while it can be a stronger or weaker condition than the classical
null energy condition. Interestingly in two dimensions QNEC$_2$ cannot be
saturated in the presence of bulk matter. The backreaction of a massive scalar
particle provides an example where the finite gap to saturation is precisely
known. Considering a massive self-interacting scalar field coupled to Einstein
gravity leads to phase transitions from small to large black holes. The dual
field theory provides a rich example to use QNEC$_2$ as a tool to learn about
strongly coupled dynamical systems. In particular knowing QNEC$_2$ in the
ground state allows us to make statements about the phase structure of the
thermal states.
