arXiv:2611.07746v2 Announce Type: replace Abstract: Volatility heredity concerns the information that past fluctuations in one series may carry about another after accounting for the target's own persistence. QS-TCN compares this dependence in both directions to estimate an order of temporal precedence. For each series, quasi-maximum likelihood estimation yields a persistence coefficient, and the matrix pencil method extracts temporal modes from the autocovariances of squared observations. These estimates define filters in a temporal convolutional network. The network models the target's squared observations from their own past and scores the association between the residuals and the filtered history of a candidate source. Differences between the two directional scores are aggregated into a global order. In an exploratory simulation with 100 systems in each of four volatility regimes, an ensemble of ten models trained with fixed seeds achieved higher mean ordering quality than CCF-AUC, a lead--lag method based on distance correlation, in every regime. Against CCF-AUC applied to squared standardized observations, the mean gains were 8.69 percentage points under positive linear coupling and 7.81 points when nonlinear coupling depended on the target's state. In the other two regimes, paired 95\% bootstrap intervals for the gains included zero. Mean agreement with reference constraints was 71.55\% across nine Finance systems and 54.02\% across 27 simulated functional magnetic resonance imaging systems, compared with 50\% expected under random ordering. In a panel of daily currency and gold returns, the estimated order placed the dollar--franc and dollar--yen exchange rates before gold.