  Recently we have proposed an unusual mechanism of superconducting current
that is specific for Quantum Hall Edge channels connected to superconducting
electrodes. We have shown that the supercurrent can be mediated by a nonlocal
electon-electon interaction that provide an opportunity for a long-distance
information transfer in the direction opposite to the electron flow. A
convenient model for such interaction is that of an external circuit. The
consideration has been done for the case of a single channel.
  In this work, we extend these results to more sophisticated setups that
include the scattering between Quantum Hall channels of opposite direction and
multiple superconducting contacts. For a single Quantum Hall constriction, we
derive a general and comprehensive relation for the interaction- induced
supercurrent in terms of scattering amplitudes and demonstrate the non-local
nature of the current by considering its sensitivity to scattering. We
understand the phase dependences of the supercurrents in multi-terminal setups
in terms of interference of Andreev reflection processes. For more complex
setups encompassing at least two constrictions we find an interplay between
non-interacting and interaction-induced currents and contibutions of complex
interference processes.
