  Whether or not the Kronecker coefficients of the symmetric group count some
set of combinatorial objects is a longstanding open question. In this work we
show that a given Kronecker coefficient is proportional to the rank of a
projector that can be measured efficiently using a quantum computer. In other
words a Kronecker coefficient counts the dimension of the vector space spanned
by the accepting witnesses of a QMA verifier, where QMA is the quantum analogue
of NP. This implies that approximating the Kronecker coefficients to within a
given relative error is not harder than a certain natural class of quantum
approximate counting problems that captures the complexity of estimating
thermal properties of quantum many-body systems. A second consequence is that
deciding positivity of Kronecker coefficients is contained in QMA,
complementing a recent NP-hardness result of Ikenmeyer, Mulmuley and Walter. We
obtain similar results for the related problem of approximating row sums of the
character table of the symmetric group. Finally, we discuss an efficient
quantum algorithm that approximates normalized Kronecker coefficients to
inverse-polynomial additive error.
