  We precisely determine the SDP value (equivalently, quantum value) of large
random instances of certain kinds of constraint satisfaction problems,
``two-eigenvalue 2CSPs''. We show this SDP value coincides with the spectral
relaxation value, possibly indicating a computational threshold. Our analysis
extends the previously resolved cases of random regular $\mathsf{2XOR}$ and
$\textsf{NAE-3SAT}$, and includes new cases such as random $\mathsf{Sort}_4$
(equivalently, $\mathsf{CHSH}$) and $\mathsf{Forrelation}$ CSPs. Our techniques
include new generalizations of the nonbacktracking operator, the Ihara--Bass
Formula, and the Friedman/Bordenave proof of Alon's Conjecture.
