We take an information complexity approach to this problem and obtain three lower bounds that improve upon earlier work. For myopic protocols (where players may see only one layer ahead but arbitrarily far behind), we greatly improve a lower bound due to Gronemeier (2006). Our new lower bound is \Omega(n/k), where n is the number of vertices per layer. For conservative protocols (where players may see arbitrarily far ahead but not behind, instead seeing only the vertex reached by following the pointers up to their layer), we extend an \Omega(n/k^2) lower bound due to Damm, Jukna and Sgall (1998) so that it applies for all k.

The above two bounds apply even to the Boolean version of pointer jumping. Our third lower bound is for the non-Boolean case and for k \leqslant log n. We obtain an \Omega(n log^(k-1) n) bound for myopic protocols. Damm et al. had obtained a similar bound for deterministic conservative protocols. All our lower bounds apply directly to randomised protocols.