  We investigate the interplay between three possible properties of stationary
point processes: i) Finite Coulomb energy with short-scale regularization, ii)
Finite $2$-Wasserstein transportation distance to the Lebesgue measure and iii)
Hyperuniformity. In dimension $2$, we prove that i) implies ii), which is known
to imply iii), and we provide simple counter-examples to both converse
implications. However, we prove that ii) implies i) for processes with a
uniformly bounded density of points, and that i) - finiteness of the
regularized Coulomb energy - is equivalent to a certain property of
quantitative hyperuniformity that is just slightly stronger than
hyperuniformity itself.
  Our proof relies on the classical link between $H^{-1}$-norm and
$2$-Wasserstein distance between measures, on the screening construction for
Coulomb gases (of which we present an adaptation to $2$-Wasserstein space which
might be of independent interest), and on recent necessary and sufficient
conditions for the existence of stationary "electric" fields compatible with a
given stationary point process.
