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We present two types of the hydrodynamic limit of the nonlinearSchr\"odinger-Chern-Simons (SCS) system. We consider two different scalings ofthe SCS system and show that each SCS system asymptotically converges towardsthe compressible and incompressible Euler system, coupled with the Chern-Simonsequations and Poisson equation respectively, as the scaled Planck constantconverges to 0. Our method is based on the modulated energy estimate. In thecase of compressible limit, we observe that the classical theory of relativeentropy method can be applied to show the hydrodynamic limit, with theadditional quantum correction term. On the other hand, for the incompressiblelimit, we directly estimate the modulated energy to derive the desiredasymptotic convergence.

Hydrodynamic limits of the nonlinear Schr\"odinger equation with the Chern-Simons gauge fields