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We prove that a sufficiently strong parallel repetition theorem for a specialcase of multiplayer (multiprover) games implies super-linear lower bounds formulti-tape Turing machines with advice. To the best of our knowledge, this isthe first connection between parallel repetition and lower bounds for timecomplexity and the first major potential implication of a parallel repetitiontheorem with more than two players.  Along the way to proving this result, we define and initiate a study of blockrigidity, a weakening of Valiant's notion of rigidity. While rigidity wasoriginally defined for matrices, or, equivalently, for (multi-output) linearfunctions, we extend and study both rigidity and block rigidity for general(multi-output) functions. Using techniques of Paul, Pippenger, Szemer\'edi andTrotter, we show that a block-rigid function cannot be computed by multi-tapeTuring machines that run in linear (or slightly super-linear) time, even in thenon-uniform setting, where the machine gets an arbitrary advice tape.  We then describe a class of multiplayer games, such that, a sufficientlystrong parallel repetition theorem for that class of games implies an explicitblock-rigid function. The games in that class have the following property thatmay be of independent interest: for every random string for the verifier(which, in particular, determines the vector of queries to the players), thereis a unique correct answer for each of the players, and the verifier accepts ifand only if all answers are correct. We refer to such games as independentgames. The theorem that we need is that parallel repetition reduces the valueof games in this class from $v$ to $v^{\Omega(n)}$, where $n$ is the number ofrepetitions.  As another application of block rigidity, we show conditional size-depthtradeoffs for boolean circuits, where the gates compute arbitrary functionsover large sets.

Block Rigidity: Strong Multiplayer Parallel Repetition implies Super-Linear Lower Bounds for Turing Machines