  We present the first analysis of top spin polarizations, $t \bar t$ spin
correlations, and $t \bar t$ spin entanglement at the LHC in the context of the
Standard Model Effective Field Theory, that goes beyond Leading Order QCD
accuracy. The complete set of independent dimension-6 operators entering $t
\bar t$ production is identified, and their effects on all top spin observables
are extracted at linear and quadratic order in $c/\Lambda^2$. By comparing
results at LO and NLO, we note that the inclusion of higher orders, while not
dramatically changing the picture, often amounts to notable numerical
differences, that are not fully captured by LO scale variation. We also find
that the expected deviations from the SM have an intricate phase space
structure, and show up predominantly at large top $p_T$. For this reason, we
advocate for the measurement of spin observables differentially or
doubly-differentially in the $t \bar t$ phase-space. We show how the inclusion
of present and future top spin measurements will improve global fits to top LHC
data, by also addressing the issue of flat directions.
