Abstract: We consider the problem of decentralized multi-agent reinforcement learning in Markov games. A fundamental question is whether there exist algorithms that, when run independently by all agents, lead to no-regret for each player, analogous to celebrated convergence results for no-regret learning in normal-form games. While recent work has shown that such algorithms exist for restricted settings (notably, when regret is defined with respect to deviations to Markov policies), the question of whether independent no-regret learning can be achieved in the standard Markov game framework was open. We provide a decisive negative resolution to this problem, both from a computational and statistical perspective. We show that: • Under the complexity-theoretic assumption that PPAD $\neq$ P, there is no polynomial-time algorithm that attains no-regret in two-player general-sum Markov games when executed independently by all players, even when the game is known to the algorithm designer. • When the game is unknown, no algorithm, efficient or otherwise, can achieve no-regret without observing exponentially many episodes in the number of players. These results are proven via lower bounds for a simpler problem we refer to as SparseCCE, in which the goal is to compute a coarse correlated equilibrium that is ``sparse'' in the sense that it can be represented as a mixture of a small number of product policies.