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Matrix scaling is a simple to state, yet widely applicable linear-algebraicproblem: the goal is to scale the rows and columns of a given non-negativematrix such that the rescaled matrix has prescribed row and column sums.Motivated by recent results on first-order quantum algorithms for matrixscaling, we investigate the possibilities for quantum speedups for classicalsecond-order algorithms, which comprise the state-of-the-art in the classicalsetting.  We first show that there can be essentially no quantum speedup in terms ofthe input size in the high-precision regime: any quantum algorithm that solvesthe matrix scaling problem for $n \times n$ matrices with at most $m$ non-zeroentries and with $\ell_2$-error $\varepsilon=\widetilde\Theta(1/m)$ must make$\widetilde\Omega(m)$ queries to the matrix, even when the success probabilityis exponentially small in $n$. Additionally, we show that for$\varepsilon\in[1/n,1/2]$, any quantum algorithm capable of producing$\frac{\varepsilon}{100}$-$\ell_1$-approximations of the row-sum vector of a(dense) normalized matrix uses $\Omega(n/\varepsilon)$ queries, and that thereexists a constant $\varepsilon_0>0$ for which this problem takes$\Omega(n^{1.5})$ queries.  To complement these results we give improved quantum algorithms in thelow-precision regime: with quantum graph sparsification and amplitudeestimation, a box-constrained Newton method can be sped up in thelarge-$\varepsilon$ regime, and outperforms previous quantum algorithms. Forentrywise-positive matrices, we find an $\varepsilon$-$\ell_1$-scaling in time$\widetilde O(n^{1.5}/\varepsilon^2)$, whereas the best previously known boundswere $\widetilde O(n^2\mathrm{polylog}(1/\varepsilon))$ (classical) and$\widetilde O(n^{1.5}/\varepsilon^3)$ (quantum).

Improved quantum lower and upper bounds for matrix scaling