Driven Quantum Stars as Controlled Primitives for Real-Time Spin Dynamics

Quantum advantage in real-time spin dynamics should be assessed against the strongest relevant classical substitutes, not merely against the qubit nature of the microscopic system. We develop a physics-based diagnostic for this boundary by reducing a qubit spin model to a spin-Landau–Lifshitz (LL) classical sector and organizing the residual quantum sector as controlled corrections. The control parameter is graph coordination: we study a spin star with $d$ leaves and $O(1\mathrm{/}d)$ hub–leaf couplings. In its homogeneous form the star benchmarks the transition from LL-substitutable dynamics to genuinely quantum, discrete-sector interference; in its fully driven form, with time-dependent fields and bilinear couplings, it is the basic message-passing primitive for tree and loopy spin structures. For coherent-state return amplitudes we prove exact leaf elimination and derive a continuous-time $1\mathrm{/}d$ hierarchy. L0 is a driven one-spin weak-mean-field theory, while G1 is a Gaussian nonlocal-in-time influence correction coupling leaf two-time kernels to the hub weak two-point function. On bounded finite-time windows away from zeros of the boundary amplitudes, the hierarchy gives $\log \mathcal{A}-\log {\mathcal{A}}_{\mathrm{L}0}=O(1\mathrm{/}d)$ and $\log \mathcal{A}-\log {\mathcal{A}}_{\mathrm{L}0}-{\mathrm{\Delta }}_{\mathrm{G}1}=O(1\mathrm{/}{d}^2)$; numerical tests on fully driven anisotropic ensembles give slopes $-1.05$ and $-2.03$. Static, inhomogeneous, aligned, and fully driven stars provide validation rungs, and comparison with a temporal matrix-product influence-matrix baseline delineates complementary regimes. Unlike rank compression on a Trotter grid, the hierarchy is ordered by a physical parameter, formulated in continuous time, and each truncation level is itself a physical theory, with the LL sector as the high-coordination limit.