How does Grover's Quantum Sorting avoid reading the list?
The list (or database) is given implicitly by an "oracle" function, which is called $\Theta(\sqrt{N})$ times throughout the algorithm. Suppose for example you're looking for a divisor of some number $N$. Then the function could map $k \leq \sqrt{N}$ to true if $k$ is a non-trivial divisor of $N$, and false otherwise. Grover's algorithm will then find a non-trivial divisor of $N$ in time $\tilde{O}(\sqrt[4]{N})$ instead of the usual $\tilde{O}(\sqrt{N})$ (note that better algorithms are available in this case, even for a classical computer).