  In this work we study quantum algorithms for Hopcroft's problem which is a
fundamental problem in computational geometry. Given $n$ points and $n$ lines
in the plane, the task is to determine whether there is a point-line incidence.
The classical complexity of this problem is well-studied, with the best known
algorithm running in $O(n^{4/3})$ time, with matching lower bounds in some
restricted settings. Our results are two different quantum algorithms with time
complexity $\widetilde O(n^{5/6})$. The first algorithm is based on partition
trees and the quantum backtracking algorithm. The second algorithm uses a
quantum walk together with a history-independent dynamic data structure for
storing line arrangement which supports efficient point location queries. In
the setting where the number of points and lines differ, the quantum walk-based
algorithm is asymptotically faster. The quantum speedups for the aforementioned
data structures may be useful for other geometric problems.
