We consider a fundamental problem in data structures, static predecessor searching: Given a subset S of size n from the universe [m], store S so that queries of the form "What is the predecessor of x in S?" can be answered efficiently. We study this problem in the cell probe model introduced by Yao. Recently, Beame and Fich obtained optimal bounds on the number of probes needed by any deterministic query scheme if the associated storage scheme uses only n^{O(1)} cells of word size (\log m)^{O(1)} bits. We give a new lower bound proof for this problem that matches the bounds of Beame and Fich. Our lower bound proof has the following advantages: it works for randomised query schemes too, while Beame and Fich's proof works for deterministic query schemes only. It also extends to `quantum address-only' query schemes that we define in this paper, and is simpler than Beame and Fich's proof. We prove our lower bound using the round elimination approach of Miltersen, Nisan, Safra and Wigderson. Using tools from information theory, we prove a strong round elimination lemma for communication complexity that enables us to obtain a tight lower bound for the predecessor problem. Our strong round elimination lemma also extends to quantum communication complexity. We also use our round elimination lemma to obtain a rounds versus communication tradeoff for the `greater-than' problem, improving on the tradeoff in Miltersen et al. We believe that our round elimination lemma is of independent interest and should have other applications.