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In this paper, we address the lower bound and space-time decay rates for thecompressible Navier-Stokes and Hall-MHD equations under $H^3-$framework in$\mathbb{R}^3$. First of all, the lower bound of decay rate for the density,velocity and magnetic field converging to the equilibrium status in $L^2$ is$(1+t)^{-\frac{3}{4}}$; the lower bound of decay rate for the first orderspatial derivative of density and velocity converging to zero in $L^2$ is$(1+t)^{-\frac{5}{4}}$, and the $k(\in [1, 3])-$th order spatial derivative ofmagnetic field converging to zero in $L^2$ is $(1+t)^{-\frac{3+2k}{4}}$.Secondly, the lower bound of decay rate for time derivatives of density andvelocity converging to zero in $L^2$ is $(1+t)^{-\frac{5}{4}}$; however, thelower bound of decay rate for time derivatives of magnetic field converging tozero in $L^2$ is $(1+t)^{-\frac{7}{4}}$. Finally, we address the decay rate ofsolution in weighted Sobolev space $H^3_{\gamma}$. More precisely, the upperbound of decay rate of the $k(\in [0, 2])$-th order spatial derivatives ofdensity and velocity converging to the $k(\in [0, 2])$-th order derivatives ofconstant equilibrium in weighted space $L^2_{\gamma}$ is$t^{-\frac{3}{4}+{\gamma}-\frac{k}{2}}$; however, the upper bounds of decayrate of the $k(\in [0, 3])$-th order spatial derivatives of magnetic fieldconverging to zero in weighted space $L^2_{\gamma}$ is$t^{-\frac{3}{4}+\frac{{\gamma}}{2}-\frac{k}{2}}$.

Lower Bound and Space-time Decay Rates of Higher Order Derivatives of Solution for the Compressible Navier-Stokes and Hall-MHD Equations