  We study a multi-terminal Josephson junction based on an interacting quantum
dot coupled to $n$ superconducting BCS leads. Using an Anderson type model of a
local level with an arbitrary onsite Coulomb repulsion, we uncover its
surprising equivalence with an effective two-terminal junction with symmetric
couplings to appropriately phase-biased leads. Regardless of the strength of
the Coulomb interaction, this hidden symmetry enables us to apply
well-established numerical and theoretical tools for exact evaluation of
various physical quantities, and imposes strict relations among them. Focusing
on three-terminal devices, we then demonstrate several phenomena such as the
existence of the finite energy band crossings, superconducting transistor and
diode effects, as well as current phase relation modulation.

Mesoscale and Nanoscale Physics;Superconductivity