Spine movements play an important role in quadrupedal locomotion, yet their potential benefits in locomotion of quadruped robots have not been systematically explored. In this work, we investigate the role of spinal joint actuation and compliance on the bounding performance of a simulated compliant quadruped robot. We designed and conducted extensive simulation experiments, to compare the benefits of different spine designs, and in particular, we compared the bounding performance when (i) using actuated versus passive spinal joint, (ii) changing the stiffness of the spinal joint and (iii) altering joint actuation profiles. We used a detailed rigid body dynamics modeling to capture the main dynamical features of the robot. We applied a set of analytic tools to evaluate the bounding gait characteristics including periodicity, stability, and cost of transport. A stochastic optimization method called particle swarm optimization was implemented to perform a global search over the parameter space, and extract a pool of diverse gait solutions. Our results show improvements in bounding speed for decreasing spine stiffness, both in the passive and the actuated case. The results also suggests that for the passive spine configuration at low stiffness values, periodic solutions are hard to realize. Overall, passive spine solutions were more energy efficient and self-stable than actuated ones, but they basically exist in limited regions of parameter space. Applying more complex joint control profiles reduced the dependency of the robot's speed to its chosen spine stiffness. In average, active spine control decreased energy efficiency and self-stability behavior, in comparison to a passive compliant spine setup.