  Bethe ansatz equations for spin-$s$ Heisenberg spin chain with $s\ge1$ are
significantly more difficult to analyze than the spin-$\tfrac{1}{2}$ case, due
to the presence of repeated roots. As a result, it is challenging to derive
extra conditions for the Bethe roots to be physical and study the related
completeness problem. In this paper, we propose the rational $Q$-system for the
XXX$_s$ spin chain. Solutions of the proposed $Q$-system give all and only
physical solutions of the Bethe ansatz equations required by completeness. This
is checked numerically and proved rigorously. The rational $Q$-system is
equivalent to the requirement that the solution and the corresponding dual
solution of the $TQ$-relation are both polynomials, which we prove rigorously.
Based on this analysis, we propose the extra conditions for solutions of the
XXX$_s$ Bethe ansatz equations to be physical.
