Infinite quantiles inference under Local Differential Privacy

We study simultaneous quantile inference under local differential privacy for a number of target probabilities that grows with the sample size. Each user participates once, so learning unknown quantile locations and calibrating their joint uncertainty must share the same private information. We fuse complementary cumulative distribution function (CDF) scores from a single tree report using their exact covariance; the variance at the correct anchor is no larger than that of either score, and anchor error incurs only quadratic excess variance. Three consecutive cohorts then localize the quantiles, estimate their densities, and supply fresh scores for a one-step correction and joint multiplier calibration. Under local density regularity, uniform identification on a public bracket, and explicit growth conditions, a uniform representation controls localization and density errors and yields asymptotically valid simultaneous coverage. A fixed-privacy regime permits every polynomially growing target grid. Simulations show consistent same-report precision gains, while comparisons with CDF-band inversion expose the sample cost of learning target locations. Census microdata illustrates how the resulting joint intervals support wage-percentile ratios.