  Shadow tomography for quantum states provides a sample efficient approach for
predicting the properties of quantum systems when the properties are restricted
to expectation values of $2$-outcome POVMs. However, these shadow tomography
procedures yield poor bounds if there are more than 2 outcomes per measurement.
In this paper, we consider a general problem of learning properties from
unknown quantum states: given an unknown $d$-dimensional quantum state $\rho$
and $M$ unknown quantum measurements $\mathcal{M}_1,...,\mathcal{M}_M$ with
$K\geq 2$ outcomes, estimating the probability distribution for applying
$\mathcal{M}_i$ on $\rho$ to within total variation distance $\epsilon$.
Compared to the special case when $K=2$, we need to learn unknown distributions
instead of values. We develop an online shadow tomography procedure that solves
this problem with high success probability requiring $\tilde{O}(K\log^2M\log
d/\epsilon^4)$ copies of $\rho$. We further prove an information-theoretic
lower bound that at least $\Omega(\min\{d^2,K+\log M\}/\epsilon^2)$ copies of
$\rho$ are required to solve this problem with high success probability. Our
shadow tomography procedure requires sample complexity with only logarithmic
dependence on $M$ and $d$ and is sample-optimal for the dependence on $K$.
