n = 1000000;
u = RandomPoint[Sphere[{0, 0, 0}], 3];

cf = Block[{u, dim},
   dim = 3;
   With[{code = KroneckerProduct[#, #] &[ Table[Compile`GetElement[u, i], {i, 1, dim}]]},
    Compile[{{u, _Real, 1}},
     code,
     CompilationTarget -> "C",
     RuntimeAttributes -> {Listable},
     Parallelization -> True,
     RuntimeOptions -> "Speed"
     ]
    ]
   ];

Q = 0.5 (Mean[cf[u]] - IdentityMatrix[3]);
p = Eigenvalues[Q, {3}][[1]]