The sumcheck protocol is a fundamental building block in the design of probabilistic proof systems, and has become a key component of recent work on efficient succinct arguments. We study time-space trade-offs for the prover of the sumcheck protocol in the streaming model, and provide upper and lower bounds that tightly characterize the efficiency achievable by the prover. We implement and evaluate the prover algorithm for products of multilinear polynomials. We show that our algorithm consumes up to \(120\times \) less memory compare to the linear-time prover algorithm, while incurring a time overhead of less than \(2\times \) . The foregoing algorithms and lower bounds apply in the interactive proof model. We show that in the polynomial interactive oracle proof model one can in fact design a new protocol that achieves a better time-space trade-off of \(O(N^{1/k})\) space and \(O(N(\log ^* N + k))\) time for any \(k \ge 1\) .