  In this letter we present constraints on the scale-dependent "local" type
primordial non-Gaussianity, which is described by non-Gaussianity's spectral
index $n_{\rm NG}$, from the NRAO VLA Sky Survey and the quasar catalog of the
Sloan Digital Sky Survey (SDSS) Data Release 6, together with the SDSS Data
Release 12 photo-z sample. Here, we use the auto-correlation analyses of these
three probes and their cross-correlation analyses with the cosmic microwave
background (CMB) temperature map, and obtain the tight constraint on the
spectral index: $n_{\rm NG}=0.2 ^{+0.7}_{-1.0}$ ($1\,\sigma$ C.L.), which shows
the first competitive constraint on the running of non-Gaussianity from current
large-scale structure clustering data. Furthermore, we also perform the
forecast calculations and improve the limit of $n_{\rm NG}$ using the future
Euclid mission, and obtain the standard deviation at 68\% confidence level:
$\Delta n_{\rm NG}=1.74$ when considering the fiducial value $f_{\rm NL}=3$,
which provides the complementary constraining power to those from the CMB
bispectrum information.
