  Using Seiberg-Witten Floer spectrum and Pin(2)-equivariant KO-theory, we
prove new Furuta-type inequalities on the intersection forms of spin cobordisms
between homology $3$-spheres. As an application, we give explicit constrains on
the intersection forms of spin $4$-manifolds bounded by Brieskorn spheres
$\pm\Sigma(2,3,6k\pm1)$. Along the way, we also give an alternative proof of
Furuta-Kametanni's improvement of 10/8-theorem for closed spin-4 manifolds.
