  We prove a family of $n$-th order identities for quantum $R$-matrices of
Baxter-Belavin type in fundamental representation. The set of identities
includes the unitarity condition as the simplest one ($n=2$). Our study is
inspired by the fact that the third order identity provides commutativity of
the Knizhnik-Zamolodchikov-Bernard connections. On the other hand the same
identity gives rise to $R$-matrix valued Lax pairs for the classical integrable
systems of Calogero type. The latter construction uses interpretation of
quantum $R$-matrix as matrix generalization of the Kronecker function. We
present a proof of the higher order scalar identities for the Kronecker
functions which is then naturally generalized to the $R$-matrix identities.
