  We present several results on the complexity of various forms of Sperner's
Lemma in the black-box model of computing. We give a deterministic algorithm
for Sperner problems over pseudo-manifolds of arbitrary dimension. The query
complexity of our algorithm is linear in the separation number of the skeleton
graph of the manifold and the size of its boundary. As a corollary we get an
$O(\sqrt{n})$ deterministic query algorithm for the black-box version of the
problem {\bf 2D-SPERNER}, a well studied member of Papadimitriou's complexity
class PPAD. This upper bound matches the $\Omega(\sqrt{n})$ deterministic lower
bound of Crescenzi and Silvestri. The tightness of this bound was not known
before. In another result we prove for the same problem an
$\Omega(\sqrt[4]{n})$ lower bound for its probabilistic, and an
$\Omega(\sqrt[8]{n})$ lower bound for its quantum query complexity, showing
that all these measures are polynomially related.
