In differential privacy (DP), we want to query a database about $n$ users, in a way that "leaks at most $\varepsilon$ about any individual user," even conditioned on any outcome of the query. Meanwhile, in gentle measurement, we want to measure $n$ quantum states, in a way that "damages the states by at most $\alpha$," even conditioned on any outcome of the measurement. In both cases, we can achieve the goal by techniques like deliberately adding noise to the outcome before returning it. This paper proves a new and general connection between the two subjects. Specifically, we show that on products of $n$ quantum states, any measurement that is $\alpha$-gentle for small $\alpha$ is also $O( \alpha)$-DP, and any product measurement that is $\varepsilon$-DP is also $O(\varepsilon\sqrt{n})$-gentle.

Illustrating the power of this connection, we apply it to the recently studied problem of shadow tomography. Given an unknown $d$-dimensional quantum state $\rho$, as well as known two-outcome measurements $E_{1},\ldots,E_{m}$, shadow tomography asks us to estimate $\Pr\left[ E_{i}\text{ accepts }\rho\right] $, for every $i\in\left[ m\right] $, by measuring few copies of $\rho$. Using our connection theorem, together with a quantum analog of the so-called private multiplicative weights algorithm of Hardt and Rothblum, we give a protocol to solve this problem using $O\left( \left( \log m\right) ^{2}\left( \log d\right) ^{2}\right)$ copies of $\rho$, compared to Aaronson's previous bound of $\widetilde{O} \left(\left( \log m\right) ^{4}\left( \log d\right)\right) $. Our protocol has the advantages of being online (that is, the $E_{i}$'s are processed one at a time), gentle, and conceptually simple.

Other applications of our connection include new lower bounds for shadow tomography from lower bounds on DP, and a result on the safe use of estimation algorithms as subroutines inside larger quantum algorithms.