Resource-Constrained Quantum Modulation of Liquid Temporal Dynamics

What functional role can a small parameterised quantum circuit play when embedded inside a classical dynamical model? We study this question through resourceconstrained quantum modulation of irregular temporal dynamics, where a lowqubit circuit modulates the effective time scale and nonlinear drift of a liquid recurrent update rather than acting as a stand-alone predictor. We characterise the induced function class and its resource sensitivity. Without entanglement, local measurements factorise into bounded nonlinear ridge functions of learned input projections; repeated data re-uploading can expand the potentially accessible local Fourier support under the stated angle-encoding assumptions, while entangling gates permit mixed-coordinate spectral interactions. We further derive a finite-shot trajectory perturbation bound showing that O(S−1/2) measurement error is filtered through temporal spacing and recurrent state sensitivity. Experiments on a controlled irregularised Room Occupancy benchmark support this mechanistic distinction. Removing entanglement while retaining re-uploading improves the classical liquid backbone from 0.3065 to 0.4018 accuracy and also improves macro-F1, negative log-likelihood, Brier score, and expected calibration error relative to the full circuit where available, whereas removing re-uploading worsens most reported metrics. Strong MLP and GRU references remain substantially better overall predictors, so the results do not support a claim of predictive quantum advantage. Finite-shot evaluation produces non-monotonic operatingpoint changes, motivating measurement budget as an explicit property of recurrent quantum models. Overall, the results support viewing low-qubit quantum circuits as structured dynamical modulators whose usefulness depends on the function class required by the downstream temporal system.