Arms races between predators and their prey are predicted to lead to reciprocal evolution producing increasingly extreme traits. I.e. "swords get sharper, so shields get thicker, so swords get sharper still" . One potential outcome of such directional selection is increased complexity [2-4]. However, phenotypes of equivalent behavioral complexity can carry different fitness effects: a grey moth with a preference for perching on grey trees is likely safer, but no more behaviorally complex, than a grey moth with a preference for perching on black trees. Given the large number of potential behavioral responses, it is unclear whether antagonistic interactions should lead to the evolution of increasingly complex behaviors or to evolutionary "chases" through behavioral strategies of comparable complexity. We examined these coevolutionary dynamics in an open-ended, evolving predator-prey system. We demonstrate that predator-prey dynamics created arms races leading to increased behavioral and information complexity, as well as intelligence. Prey evolved with predators had a higher intake of sensory information about their environment and used more of that information in decision making than those evolved in the absence of predators. Moreover, prey that evolved with predators conditionally responded more to information about food, allowing for more adept foraging. While hunting and avoidance strategies continued to improve over evolutionary time, observed complexity plateaued. Thus we show that while the escalation inherent to arms races can produce complex traits, including basic intelligence, further significant evolutionary advancements arise from concurrent non-escalating chases through options in behavioral space.