Mesoscale simulation approaches that leverage the scaling of atomic potential functions, such as quasi-coarse-grained dynamics (QCGD), have demonstrated considerable utility in modeling crystalline systems. This study systematically analyzes the relationship between QCGD and molecular dynamics (MD) to reveal a strict spatiotemporal conjugacy between the two methodologies. This theoretical foundation confirms that QCGD effectively amplifies MD trajectories while preserving key structural and thermodynamic properties, providing a rigorous basis for concurrent multi-scale modeling. Building on this conjugacy, we propose a novel concurrent QCGD-MD coupling framework designed to bridge the gap between microscopic fidelity and mesoscopic efficiency. Unlike traditional methods that rely on non-conserving "handshaking" regions, our approach employs a direct interface interaction strategy within a unified Hamiltonian framework. This design philosophy ensures that the coupled system intrinsically satisfies the global conservation of energy and momentum, which significantly enhances numerical stability during long-term dynamic simulations involving complex deformations. To address the thermodynamic challenges arising from the scale disparity, particularly the severe phonon spectrum mismatch between coarse and fine particles, we introduce a dual-correction mechanism involving adaptive heat flux control and energy-conserving Dissipative Particle Dynamics (DPD-E). Conceptually, the DPD-E layer functions as a thermalizing buffer that effectively dissipates high-frequency phonons and ballistic waves reflected or transmitted at the interface. By converting coherent wave energy into thermal motion, this mechanism suppresses spurious interfacial heating and ensures a smooth thermodynamic transition across scales without violating conservation laws. By harmonizing the mechanical consistency of the Hamiltonian framework with the thermodynamic consistency provided by DPD-E, this work establishes a robust and physically rigorous paradigm for cross-scale simulations of dynamic processes in materials.