  Starting with a set of conditions for bipartite separability of arbitrary
quantum states in any dimension and expressed in terms of arbitrary operators
whose commutator is a $c$-number, we derive a hierarchy of conditions for
tripartite separability of continuous-variable three-mode quantum states. These
conditions have the form of inequalities for higher-order moments of linear
combinations of the mode operators. They enable one to distinguish between all
possible kinds of tripartite separability, while the strongest violation of
these inequalities is a sufficient condition for genuine tripartite
entanglement. We construct Gaussian states for which the violation of our
conditions grows exponentially with the order of the moments of the mode
operators. By going beyond second moments, our conditions are expected to be
useful as well for the detection of tripartite entanglement of non-Gaussian
states.
