  In this paper, we are concerned with the energy and magnetic helicity
conservation of weak solutions for both the electron and Hall
magnetohydrodynamic equations. Various sufficient criteria to ensure the energy
and magnetic helicity conservation in Onsager's critical spaces
$\underline{B}^{\alpha}_{p,VMO}$ and $B^{\alpha}_{p,c(\mathbb{N})}$ in these
systems are established. Moreover, for the E-MHD equations, we observe that the
conservation criteria of energy and magnetic helicity to the E-MHD equations
correspond to the helicity and energy to the ideal incompressible Euler
equations, respectively.
