Sensitive statistics are often collected across sets of users, with repeated collection of reports done over time. For example, trends in users private preferences or software usage may be monitored via such reports. We study the collection of such statistics in the local differential privacy (LDP) model, and describe an algorithm whose privacy cost is polylogarithmic in the number of changes to a users value. More fundamentally---by building on anonymity of the users reports---we also demonstrate how the privacy cost of our LDP algorithm can actually be much lower when viewed in the central model of differential privacy. We show, via a new and general privacy amplification technique, that any permutation-invariant algorithm satisfying $varepsilon$-local differential privacy will satisfy $(O(varepsilon sqrt{log(1/delta)/n}), delta)$-central differential privacy. By this, we explain how the high noise and $sqrt{n}$ overhead of LDP protocols is a consequence of them being significantly more private in the central model. As a practical corollary, our results imply that several LDP-based industrial deployments may have much lower privacy cost than their advertised $varepsilon$ would indicate---at least if reports are anonymized.
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