  In this paper, we address the lower bound and space-time decay rates for the
compressible 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 order
spatial 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 of
magnetic 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 and
velocity converging to zero in $L^2$ is $(1+t)^{-\frac{5}{4}}$; however, the
lower bound of decay rate for time derivatives of magnetic field converging to
zero in $L^2$ is $(1+t)^{-\frac{7}{4}}$. Finally, we address the decay rate of
solution in weighted Sobolev space $H^3_{\gamma}$. More precisely, the upper
bound of decay rate of the $k(\in [0, 2])$-th order spatial derivatives of
density and velocity converging to the $k(\in [0, 2])$-th order derivatives of
constant equilibrium in weighted space $L^2_{\gamma}$ is
$t^{-\frac{3}{4}+{\gamma}-\frac{k}{2}}$; however, the upper bounds of decay
rate of the $k(\in [0, 3])$-th order spatial derivatives of magnetic field
converging to zero in weighted space $L^2_{\gamma}$ is
$t^{-\frac{3}{4}+\frac{{\gamma}}{2}-\frac{k}{2}}$.
