  Cosmological inflation predicts that the scalar spectral index "runs" with
scale. Constraints on the values of the spectral runnings, $\alpha_s\equiv
\textrm{d} n_s/\textrm{d}\ln k$ and $\beta_s\equiv
\textrm{d}\alpha_s/\textrm{d}\ln k$, therefore provide a fundamental test of
the physics of inflation. Here we study the feasibility of measuring the
runnings when information from upcoming large-volume galaxy surveys is used to
supplement the information provided by a CMB-S4 experiment, particularly
focusing on the effect of including high-$k$, nonlinear scales. Since these
measurements will be sensitive to modeling uncertainties for the nonlinear
power spectrum, we examine how three different ways of parameterizing those
systematics---introducing zero, two, or several hundred nuisance
parameters---affect constraints and protect against parameter biases.
Considering statistical errors alone, we find that including strongly nonlinear
scales can substantially tighten constraints. However, these constraints weaken
to levels not much better than those from a CMB-S4 experiment alone when we
limit our analysis to scales where estimates are not strongly affected by
systematic biases. Given these considerations, near-future large-scale
structure surveys are unlikely to add much information to the CMB-S4
measurement of the first running $\alpha_s$. There is more potential for
improvement for the second running, $\beta_s$, for which large-scale structure
information will allow constraints to be improved by a factor of 3--4 relative
to using the CMB alone. Though these constraints are still above the value
predicted by slow roll inflation, they do probe regions of parameter space
relevant to nonstandard inflationary models with large runnings, for example
those that can generate an appreciable abundance of primordial black holes.
