The reachability problem for timed automata asks if a given automaton has a run leading to an accepting state, and the liveness problem asks if the automaton has an infinite run which visits accepting states infinitely often. Both these problems are known to be Pspace-complete. We show that if P Pspace, the liveness problem is more difficult than the reachability problem; in other words we exhibit a family of automata for which solving the reachability problem with the standard algorithm is in P but solving the liveness problem is Pspace-hard. This leads us to revisit the algorithmics for the liveness problem. We propose a notion of a witness for the fact that a timed automaton violates a liveness property. We give an algorithm for computing such a witness and compare it with existing solutions. CCS Concepts: • Theory of computation → Verification by model checking.