  We conjecture a new way to construct eigenstates of integrable XXX quantum
spin chains with SU(N) symmetry. The states are built by repeatedly acting on
the vacuum with a single operator Bgood(u) evaluated at the Bethe roots. Our
proposal serves as a compact alternative to the usual nested algebraic Bethe
ansatz. Furthermore, the roots of this operator give the separated variables of
the model, explicitly generalizing Sklyanin's approach to the SU(N) case. We
present many tests of the conjecture and prove it in several special cases. We
focus on rational spin chains with fundamental representation at each site, but
expect many of the results to be valid more generally.
