  Witten's proof for the positivity of the ADM mass gives a definition of
energy in terms of three-surface spinors. In this paper, we give a
generalisation for the remaining six Poincar\'e charges at spacelike infinity,
which are the angular momentum and centre of mass. The construction improves on
certain three-surface spinor equations introduced by Shaw. We solve these
equations asymptotically obtaining the ten Poincar\'e charges as integrals over
the Nester--Witten two-form. We point out that the defining differential
equations can be extended to three-surfaces of arbitrary signature and we study
them on the entire boundary of a compact four-dimensional region of spacetime.
The resulting quasi-local expressions for energy and angular momentum are
integrals over a two-dimensional cross-section of the boundary. For any two
consecutive such cross-sections, conservation laws are derived that determine
the influx (outflow) of matter and gravitational radiation.
