Quantum HS°: A Resource Theory of Temporal Alignment v1.1

We define Quantum HS$^\circ$ (Harmony Segment Degree) as a minimal quantum resource theory for temporal (phase) alignment when no external synchronized clock is available. The physical restriction is captured by a $U(1)$ time/phase-translation symmetry: free operations are time-translation covariantquantum channels, and free states are time-translation invariant states. As a canonical monotone, we use the relative-entropy of temporal asymmetry$A_{\mathrm{HS}}(\rho)=S(\mathcal{T}(\rho))-S(\rho)$, where $\mathcal{T}$ is the group-twirling (symmetry-averaging) channel. We show (i) equivalence to therelative-entropy distance to the invariant set, (ii) monotonicity under covariant channels, and (iii) an operational interpretation: for a uniform unknown phaseoffset encoding ensemble, $A_{\mathrm{HS}}(\rho)$ equals the Holevo information.This frames temporal asymmetry as an operational and consumable resource foralignment tasks without invoking time operators or thermodynamic assumptions.