  In this note, we investigate the implications of classical soft theorems for
the formalism developed by Kosower, Maybee and O'Connell (KMOC) to derive
classical observables in gauge theory and gravity from scattering amplitudes.
In particular, we show that the radiative electro-magnetic field at leading
order in the soft expansion imposes an infinite hierarchy of constraints on the
expectation value of the family of observables generated by \textit{monomials}
of linear impulse. We perform an explicit check on these constraints at next to
leading order (NLO) in the coupling and as a corollary show how up to NLO, soft
radiation obtained from quantum amplitudes is consistent with the (leading)
classical soft photon theorem.
  We also argue that in 4 dimensions the classical log soft theorem derived by
Saha, Sahoo and Sen generates an infinite hierarchy of constraints on the
expectation value of operators which are products of one angular momentum and
an arbitrary number of linear momenta.
