  We give a microscopic derivation of time-dependent correlation functions of
the $1D$ cubic nonlinear Schr\"{o}dinger equation (NLS) from many-body quantum
theory. The starting point of our proof is our previous work on the
time-independent problem and work of the second author on the corresponding
problem on a finite lattice. An important new obstacle in our analysis is the
need to work with a cutoff in the number of particles, which breaks the
Gaussian structure of the free quantum field and prevents the use of the Wick
theorem. We overcome it by the means of complex analytic methods. Our methods
apply to the nonlocal NLS with bounded convolution potential. In the periodic
setting, we also consider the local NLS, arising from short-range interactions
in the many-body setting. To that end, we need the dispersion of the NLS in the
form of periodic Strichartz estimates in $X^{s,b}$ spaces.
