  We discuss to what extent the full set of Ward Identities constrain the
four-point function of the stress-energy tensors or conserved currents in a
conformal field theory. We calculate the number of kinematically unrestricted
functional degrees of freedom governing the corresponding correlators and find
that it matches the number of functional degrees of freedom governing
scattering amplitudes of some "dual" massless particles in the auxiliary
Minkowski space. We also formulate the conformal bootstrap constraints for the
correlators in question in terms of only unrestricted degrees of freedom. As a
by-product we find interesting parallels between solving Ward Identities in
coordinate and momentum space.
