Assume we are given a quantum field theory described by some functional. If $J$ is a Noether current, i.e. it is associated with a symmetry of the functional and satisfies $\partial_s J^s=0$ (Noether theorem), we can always obtain a conserved quantity as $Q(x^0) = \int J^0(x^0, x^1,...,x^n)d^{n}x$. What is the converse to the previous statement? I would like to know what assumptions are necessary to build a conserved current $J$ given $Q$ swith $\dot Q = [H,Q]=0$ where $H$ is the Hamiltonian of the (quantum field) theory.