  Phase-space representations as given by Wigner functions are a powerful tool
for representing the quantum state and characterizing its time evolution in the
case of infinite-dimensional quantum systems and have been widely used in
quantum optics and beyond. Continuous phase spaces have also been studied for
finite-dimensional quantum systems such as spin systems. However, much less is
known for finite-dimensional, coupled systems, and we present a complete theory
of Wigner functions for this case. In particular, we provide a self-contained
Wigner formalism for describing and predicting the time evolution of coupled
spins which lends itself to visualizing the high-dimensional structure of
multi-partite quantum states. We completely treat the case of an arbitrary
number of coupled spins 1/2, thereby establishing the equation of motion using
Wigner functions. The explicit form of the time evolution is then calculated
for up to three spins 1/2. The underlying physical principles of our Wigner
representations for coupled spin systems are illustrated with multiple examples
which are easily translatable to other experimental scenarios.
