Bilinear higher-spin currents in the unfolded formalism

The classification of primary non-trivial bilinear currents in the 4d higher-spin theory is obtained. It is interpreted in terms of the known classification of cubic Lagrangian vertices in the 4d higher-spin theory. It is shown that some currents join to the action with the tail of higher-derivative terms (as it happens with the Fradkin-Vasiliev $2-s-s$ vertex). The analysis is based on the $\sigma_-$-cohomology technique of the unfolded formalism.