We present singly-exponential quantum algorithms for the One-Sided Crossing Minimization (OSCM) problem. Given an (n)-vertex bipartite graph (G=(U,V,E\subseteq U \times V)), a (2)-level drawing ((\pi_U,\pi_V)) of (G) is described by a linear ordering (\pi_U: U \leftrightarrow \{1,\dots,|U|\}) of (U) and linear ordering (\pi_V: V \leftrightarrow \{1,\dots,|V|\}) of (V). For a fixed linear ordering (\pi_U) of (U), the OSCM problem seeks to find a linear ordering (\pi_V) of (V) that yields a (2)-level drawing ((\pi_U,\pi_V)) of (G) with the minimum number of edge crossings. We show that OSCM can be viewed as a set problem over (V) amenable for exact algorithms with a quantum speedup with respect to their classical counterparts. First, we exploit the quantum dynamic programming framework of Ambainis et al. [Quantum Speedups for Exponential-Time Dynamic Programming Algorithms. SODA 2019] to devise a QRAM-based algorithm that solves OSCM in (O^(1.728^n)) time and space. Second, we use quantum divide and conquer to obtain an algorithm that solves OSCM without using QRAM in (O^(2^n)) time and polynomial space.