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This talk deals with versions of Rudyak’s extension of Farber’s topological complexity.We describe a (conjecturally sharp) upper bound for the higher symmetric topological complexity of spheres.Methods depend on a reasonable understanding of the homotopical properties of configuration spaces of spheres.Here the concept of a cellular stratified space is central,as it allows us to import techniques from the theory of hyperplane arrangements in order to construct finite simplicial complexes of the lowest possible dimension modeling,up to Σn-equivariant homotopy,configuration spaces of n ordered distinct points on spheres.