The Material Point Method (MPM) is a numerical framework capable of handling large deformations by representing material behavior through particles embedded in a background computational grid. Owing to this feature, MPM is well suited for geotechnical problems involving extremely large deformation. However, its application to dynamic problems, such as earthquake-induced ground response, still faces several technical challenges. This study presents two independent methodological developments aimed at extending the capability of MPM for dynamic analyses. The first development is on a two-phase MPM formulation for liquefaction analysis. A single-point u-v-p formulation is adopted to model the coupled solid-fluid behavior of saturated soils. A semi-implicit time integration scheme is employed to decouple the pore-pressure field from the kinematic variables, thereby improving numerical stability. Sand behavior under cyclic loading is described using a bounding surface plasticity model. The proposed framework is validated against centrifuge experiments conducted on Toyoura sand, demonstrating good agreement in terms of excess pore pressure generation as well as large-deformation responses, including embankment settlement and lateral spreading. The second development focuses on an enhanced Perfectly Matched Layer (PML) formulation for MPM. Absorbing boundary conditions are essential for realistic simulation of in-situ problems, as outgoing waves must be allowed to leave the computational domain without artificial reflection, thereby reducing the required analysis domain. In our previous work, PML was implemented within MPM using a simplified formulation. In the present study, a more rigorous PML formulation is introduced to achieve higher wave attenuation rates. Numerical results demonstrate improved absorption performance while maintaining computational stability. These two developments are currently pursued independently. Future work will focus on integrating the proposed two-phase formulation with the enhanced PML framework to enable comprehensive simulation of earthquake-induced large-deformation problems, including liquefaction effects.