Title: Successor Uncertainties: exploration and uncertainty in temporal difference learning. Abstract: We consider the problem of balancing exploration and exploitation in sequential decision making problems. This trade-off naturally lends itself to probabilistic modelling. For a probabilistic approach to be effective, considering uncertainty about all immediate and long-term consequences of agent's actions is vital. An estimate of such uncertainty can be leveraged to guide exploration even in situations where the agent needs to perform a potentially long sequence of actions before reaching an under-explored area of the environment. This observation was made by the authors of the Uncertainty Bellman Equation model (O'Donoghue et al., 2018), which explicitly considers full marginal uncertainty for each decision the agent faces. However, their model still considers a fully factorised posterior over the consequences of each action, meaning that dependencies vital for correlated long-term exploration are ignored. We go a step beyond and develop Successor Uncertainties, a probabilistic model for the state-action value function of a Markov Decision Process with a non-factorised covariance. We demonstrate how this leads to greatly improved performance on classic tabular exploration benchmarks and show strong performance of our method on a subset of Atari baselines. Overall, Successor Uncertainties provides a better probabilistic model for temporal difference learning at a similar computational cost to its predecessors.