  We present two types of the hydrodynamic limit of the nonlinear
Schr\"odinger-Chern-Simons (SCS) system. We consider two different scalings of
the SCS system and show that each SCS system asymptotically converges towards
the compressible and incompressible Euler system, coupled with the Chern-Simons
equations and Poisson equation respectively, as the scaled Planck constant
converges to 0. Our method is based on the modulated energy estimate. In the
case of compressible limit, we observe that the classical theory of relative
entropy method can be applied to show the hydrodynamic limit, with the
additional quantum correction term. On the other hand, for the incompressible
limit, we directly estimate the modulated energy to derive the desired
asymptotic convergence.
