In this work, we propose a new way to (non-interactively, verifiably) demonstrate quantum advantage by solving the average-case $\mathsf{NP}$ search problem of finding a solution to a system of (underdetermined) constant degree multivariate equations over the finite field $\mathbb{F}_2$ drawn from a specified distribution. In particular, for any $d \geq 2$, we design a distribution of degree up to $d$ polynomials $\{p_i(x_1,\ldots,x_n)\}_{i\in [m]}$ for $m<n$ over $\mathbb{F}_2$ for which we show that there is a expected polynomial-time quantum algorithm that provably simultaneously solves $\{p_i(x_1,\ldots,x_n)=y_i\}_{i\in [m]}$ for a random vector $(y_1,\ldots,y_m)$. On the other hand, while solutions exist with high probability, we conjecture that for constant $d > 2$, it is classically hard to find one based on a thorough review of existing classical cryptanalysis. Our work thus posits that degree three functions are enough to instantiate the random oracle to obtain non-relativized quantum advantage. Our approach begins with the breakthrough Yamakawa-Zhandry (FOCS 2022) quantum algorithmic framework. In our work, we demonstrate that this quantum algorithmic framework extends to the setting of multivariate polynomial systems. Our key technical contribution is a new analysis on the Fourier spectra of distributions induced by a general family of distributions over $\mathbb{F}_2$ multivariate polynomials -- those that satisfy $2$-wise independence and shift-invariance. This family of distributions includes the distribution of uniform random degree at most $d$ polynomials for any constant $d \geq 2$. Our analysis opens up potentially new directions for quantum cryptanalysis of other multivariate systems.