  We introduce a notion of complexity for systems of linear forms, called
sequential Cauchy-Schwarz complexity, which is parametrized by two positive
integers $k,\ell$ and refines the notion of Cauchy-Schwarz complexity
introduced by Green and Tao. We prove that if a system of linear forms has
sequential Cauchy-Schwarz complexity at most $(k,\ell)$ then any average of
1-bounded functions over this system is controlled by the $2^{1-\ell}$-th power
of the Gowers $U^{k+1}$-norms of the functions. For $\ell=1$ this agrees with
Cauchy-Schwarz complexity, but for $\ell>1$ there are systems that have
sequential Cauchy-Schwarz complexity at most $(k,\ell)$ whereas their
Cauchy-Schwarz complexity is greater than $k$. Our main application illustrates
this with systems over a prime field $\mathbb{F}_p$ that are denoted by
$\Phi_{k,M}$ and can be viewed as $M$-dimensional arithmetic progressions of
length $k$. For each $M\geq 2$ we prove that $\Phi_{k,M}$ has sequential
Cauchy-Schwarz complexity at most $(k-2,|\Phi_{k,M}|)$ (where $|\Phi_{k,M}|$ is
the number of forms in the system), whereas the Cauchy-Schwarz complexity of
$\Phi_{k,M}$ can be greater than $k-2$. Thus we obtain polynomial
true-complexity bounds for $\Phi_{k,M}$ with exponent $2^{-|\Phi_{k,M}|}$. A
recent general theorem of Manners, proved independently with different methods,
implies a similar application but with different polynomial true-complexity
bounds, as explained in the paper. In separate work, we use our application to
give a new proof of the inverse theorem for Gowers norms on $\mathbb{F}_p^n$,
and related results concerning ergodic actions of $\mathbb{F}_p^{\omega}$.
