  We review some recent results on Sorkin's spacetime formulation of the
entanglement entropy (SSEE) for a free quantum scalar field both in the
continuum and in manifold-like causal sets. The SSEE for a causal diamond in a
2d cylinder spacetime has been shown to have a Calabrese-Cardy form, while for
de Sitter and Schwarzschild de Sitter horizons in dimensions $d>2$, it matches
the mode-wise von-Neumann entropy. In these continuum examples the SSEE is
regulated by imposing a UV cut-off. Manifold-like causal sets come with a
natural covariant spacetime cut-off and thus provide an arena to study
regulated QFT. However, the SSEE for different manifold like causal sets in
$d=2$ and $d=4$ has been shown to exhibit a volume rather than an area law. The
area law is recovered only when an additional UV cut-off is implemented in the
scaling regime of the spectrum which mimics the continuum behaviour. We discuss
the implications of these results and suggest that a volume-law may be a
manifestation of the fundamental non-locality of causal sets and a sign of new
UV physics.
