  Collinear laser spectroscopy is performed on the nickel isotopes
$^{58-68,70}$Ni, using a time-resolved photon counting system. From the
measured isotope shifts, nuclear charge radii $R_c$ are extracted and compared
to theoretical results. Three ab initio approaches all employ, among others,
the chiral interaction NNLO$_{\rm sat}$, which allows an assessment of their
accuracy. We find agreement with experiment in differential radii $\delta
\left\langle r_\mathrm{c}^2 \right\rangle$ for all employed ab initio methods
and interactions, while the absolute radii are consistent with data only for
NNLO$_{\rm sat}$. Within nuclear density functional theory, the Skyrme
functional SV-min matches experiment more closely than the Fayans functional
Fy($\Delta r$,HFB).
