We show two unconditional quantum space advantages in the random-order streaming model. First, we show the Yamakawa-Zhandry Code Intersection problem admits exponential quantum advantage in the streaming model when its inputs are streamed in random order. This means quantum computers exhibit exponential space advantage even when simply receiving $(x,f(x))$ pairs for a uniformly random function $f$ in a uniformly random order. Notably, our classical space lower bound does not follow from a communication lower bound. Instead, we utilize density-restoring partitions from communication complexity directly with potential-function analysis in the style used to prove memory-sample tradeoffs for learning. The resulting argument is distinct from both lines of work. Second, we use our framework to show quantum space advantage for the Optimal Polynomial Intersection (OPI) problem in certain regimes via a streaming version of the Decoded Quantum Interferometry algorithm. In particular, we show that for degree $d$ and $n$ evaluation points, attaining $1/2 + Ω(\sqrt{d/n})$ fraction of satisfied OPI constraints via streaming requires $Ω(n)$ classical bits of memory but only $O(d\log n)$ qubits. This yields provable quantum advantage in a "low-rate" regime when the number of evaluation points is much larger than the degree, with a space advantage that can be as large as exponential in certain parameter settings.

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