  We prove the expected disturbance caused to a quantum system by a sequence of
randomly ordered two-outcome projective measurements is upper bounded by the
square root of the probability that at least one measurement in the sequence
accepts. We call this bound the Gentle Random Measurement Lemma.
  We then consider problems in which we are given sample access to an unknown
state $\rho$ and asked to estimate properties of the accepting probabilities
$\text{Tr}[M_i \rho]$ of a set of measurements $\{M_1, M_2, \ldots , M_m\}$. We
call these types of problems Quantum Event Learning Problems. Using the gentle
random measurement lemma, we show randomly ordering projective measurements
solves the Quantum OR problem, answering an open question of Aaronson. We also
give a Quantum OR protocol which works on non-projective measurements but which
requires a more complicated type of measurement, which we call a Blended
Measurement. Given additional guarantees on the set of measurements $\{M_1,
\ldots, M_m\}$, we show the Quantum OR protocols developed in this paper can
also be used to find a measurement $M_i$ such that $\text{Tr}[M_i \rho]$ is
large. We also give a blended measurement based protocol for estimating the
average accepting probability of a set of measurements on an unknown state.
  Finally we consider the Threshold Search Problem described by O'Donnell and
B\u{a}descu. By building on our Quantum Event Finding result we show that
randomly ordered (or blended) measurements can be used to solve this problem
using $O(\log^2(m) / \epsilon^2)$ copies of $\rho$. Consequently, we obtain an
algorithm for Shadow Tomography which requires
$\tilde{O}(\log^2(m)\log(d)/\epsilon^4)$ samples, matching the current best
known sample complexity. This algorithm does not require injected noise in the
quantum measurements, but does require measurements to be made in a random
order and so is no longer online.
