We study the query complexity of estimating scalar summaries of zigzag bar lifetimes from snapshot-adjacency bits. For graphs G_1,\ldots,G_m on n labeled vertices, let \ell_b be the snapshot lifetime of a degree-one bar b of the intersection
We study the query complexity of estimating scalar summaries of zigzag bar lifetimes from snapshot-adjacency bits. For graphs G_1,\ldots,G_m on n labeled vertices, let \ell_b be the snapshot lifetime of a degree-one bar b of the intersection zigzag. For a probability generating function \phi(x)=E[x^R], the statistic F_\phi=\sum_b\phi(\ell_b/m) includes normalized degree-r total persistence and the mean generalized rank over a uniform time window. An exact identity underlies our algorithm: sample R uniform times; the expected generalized rank between their minimum and maximum equals F_\phi. For graphs that rank is the circuit rank of an intersection graph, so a nonlinear barcode functional becomes an average of edge and component counts, and no barcode is computed. Without spectral-gap, homology-state, or QRAM assumptions, this gives a quantum estimator with additive error \varepsilon n and \widetilde O(\sqrt{m(K+n)}/\varepsilon) queries when a bound K\ge F_\phi is supplied, against \widetilde O(m\min\{n^2,(K+n)/\varepsilon^2\}) classically, and an adaptive quantum variant with the same instance dependence. These estimators are optimal in two regimes. For every fixed power weight x^r, r\ge2, and for the uniform-window mean, the worst-case complexities are \widetilde\Theta(n\sqrt m/\varepsilon) quantum and \Theta(n^2m) classical. On sparse instances, under an explicit split-leakage promise met by power and binomial weights of logarithmic degree and the promise F_\phi\le K, they are \widetilde\Theta(\sqrt{mK}/\varepsilon) and \widetilde\Theta(m\min\{n^2,K/\varepsilon^2\}). The classical lower bounds hold against fully adaptive algorithms, and fewer than m such statistics cannot determine the positive-lifetime histogram. All bounds concern snapshot access; with an explicit update stream, near-linear full-barcode algorithms are known.