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. For sumcheck claims about a single multilinear polynomial we demonstrate an algorithm that runs in time O(kN) and uses space O(N1/k) for any k≥1. For non-adaptive provers (a class which contains all known sumcheck prover algorithms) we show this trade-off is optimal.For sumcheck claims about products of multilinear polynomials, we describe a prover algorithm that runs in time O(N(loglogN+k)) and uses space O(N1/k) for any k≥1. We show that, conditioned on the hardness of a natural problem about multiplication of multilinear polynomials, any “natural” prover algorithm that uses space O(N1/2-ε) for some ε>0 must run in time Ω(N(loglogN+logε)). For sumcheck claims about a single multilinear polynomial we demonstrate an algorithm that runs in time O(kN) and uses space O(N1/k) for any k≥1. For non-adaptive provers (a class which contains all known sumcheck prover algorithms) we show this trade-off is optimal. For sumcheck claims about products of multilinear polynomials, we describe a prover algorithm that runs in time O(N(loglogN+k)) and uses space O(N1/k) for any k≥1. We show that, conditioned on the hardness of a natural problem about multiplication of multilinear polynomials, any “natural” prover algorithm that uses space O(N1/2-ε) for some ε>0 must run in time Ω(N(loglogN+logε)). We implement and evaluate the prover algorithm for products of multilinear polynomials. We show that our algorithm consumes up to 120× less memory compare to the linear-time prover algorithm, while incurring a time overhead of less than 2×. 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(N1/k) space and O(N(log∗N+k)) time for any k≥1.