Modern collaborative robotic applications require robot motions that are predictable for human coworkers. Therefore, trajectories often need to be planned in task space rather than configuration space ($\mathcal{C}$-space). While the interpolation of translations in Euclidean space is straightforward, the interpolation of rotations in$SO(3)$is more complex. Most approaches originating from computer graphics do not exhibit the often desired$\mathcal{C}^{2}$-continuity in robotics. Our main contribution is a$\mathcal{C}^{2}$-continuous, zero-clamped interpolation scheme for quaternions that computes a fast synchronized motion given a set of waypoints. As a second contribution, we present modifications to two state-of-the-art quaternion interpolation schemes, Spherical Quadrangle Interpolation (SQUAD) and Spherical Parabolic Blends (SPB), to enable them to compute$\mathcal{C}^{2}$-continuous, zero-clamped trajectories. In experiments, we demonstrate that for the time optimization of trajectories, our approach is computationally efficient and at the same time computes smooth trajectory profiles.