  The classical 3SUM conjecture states that the class of 3SUM-hard problems
does not admit a truly subquadratic $O(n^{2-\delta})$-time algorithm, where
$\delta >0$, in classical computing. The geometric 3SUM-hard problems have
widely been studied in computational geometry and recently, these problems have
been examined under the quantum computing model. For example, Ambainis and
Larka [TQC'20] designed a quantum algorithm that can solve many geometric
3SUM-hard problems in $O(n^{1+o(1)})$-time, whereas Buhrman [ITCS'22]
investigated lower bounds under quantum 3SUM conjecture that claims there does
not exist any sublinear $O(n^{1-\delta})$-time quantum algorithm for the 3SUM
problem. The main idea of Ambainis and Larka is to formulate a 3SUM-hard
problem as a search problem, where one needs to find a point with a certain
property over a set of regions determined by a line arrangement in the plane.
The quantum speed-up then comes from the application of the well-known quantum
search technique called Grover search over all regions.
  This paper further generalizes the technique of Ambainis and Larka for some
3SUM-hard problems when a solution may not necessarily correspond to a single
point or the search regions do not immediately correspond to the subdivision
determined by a line arrangement. Given a set of $n$ points and a positive
number $q$, we design $O(n^{1+o(1)})$-time quantum algorithms to determine
whether there exists a triangle among these points with an area at most $q$ or
a unit disk that contains at least $q$ points. We also give an
$O(n^{1+o(1)})$-time quantum algorithm to determine whether a given set of
intervals can be translated so that it becomes contained in another set of
given intervals and discuss further generalizations.
