  The entanglement entropy in many gapless quantum systems receives a
contribution from corners in the entangling surface in 2+1d. It is
characterized by a universal function $a(\theta)$ depending on the opening
angle $\theta$, and contains pertinent low energy information. For conformal
field theories (CFTs), the leading expansion coefficient in the smooth limit
$\theta \to \pi$ yields the stress tensor 2-point function coefficient $C_T$ .
Little is known about $a(\theta)$ beyond that limit. Here, we show that the
next term in the smooth limit expansion contains information beyond the 2- and
3-point correlators of the stress tensor. We conjecture that it encodes 4-point
data, making it much richer. Further, we establish strong constraints on this
and higher order smooth-limit coefficients. We also show that $a(\theta)$ is
lower-bounded by a non-trivial function multiplied by the central charge $C_T$
, e.g. $a(\pi/2) \geq (\pi^2 \ln 2)C_T /6$. This bound for 90-degree corners is
nearly saturated by all known results, including recent numerics for the
interacting Wilson-Fisher quantum critical points (QCPs). A bound is also given
for the R\'enyi entropies. We illustrate our findings using O(N) QCPs, free
boson and Dirac fermion CFTs, strongly coupled holographic ones, and other
models. Exact results are also given for Lifshitz quantum critical points, and
for conical singularities in 3+1d.
