  To study properties of magneto-hydrodynamic (MHD) convection and resultant
dynamo activities in proto-neutron stars (PNSs), we construct a "PNS in a box"
simulation model with solving compressible MHD equation coupled with a nuclear
equation of state (EOS) and a simplified leptonic transport. As a
demonstration, we apply it to two types of PNS models with different internal
structures: fully-convective model and spherical-shell convection model. By
varying the spin rate of models, the rotational dependence of convection and
dynamo that operate inside the PNS is investigated. We find that, as a
consequence of turbulent transport by rotating stratified convection,
large-scale structures of flow and thermodynamic fields are developed in all
models. Depending on the spin rate and the convection zone depth, various
profiles of the large-scale structures are obtained, which can be physically
understood as steady-state solutions to the "mean-field" equation of motion.
Additionally to those hydrodynamic structures, the large-scale magnetic
component with $\mathcal{O}(10^{15})$ G is also spontaneously organized in
disordered tangled magnetic fields in all models. The higher the spin rate, the
stronger the large-scale magnetic component is built up. Intriguingly, as an
overall trend, the fully-convective models have a stronger large-scale magnetic
component than that in the spherical-shell convection models. The deeper the
convection zone extends, the larger the size of the convection eddies becomes.
As a result, the rotationally-constrained convection seems to be more easily
achieved in the fully-convective model, resulting in the higher efficiency of
the large-scale dynamo there. To gain a better understanding of the origin of
the diversity of NS's magnetic field, we need to study the PNS dynamo in a
wider parameter range.
