The non-uniform versions of P, NP and coNP are P/poly, NP/poly and coNP/poly. Similarly, we can define a non-uniform version for each level in the PH.
For example: $\Sigma_2$/poly consists of problems of the form $\{x : \exists y \forall z \; C(x,y,z)\}$, where C is a circuit of polynomial size that may vary depending on the length of the input string $x$, and $y,z$ also have lengths polynomial in $x$.
Doing this for all levels of PH, we get a non-uniform version PH/poly.
QUESTIONS: Is there anything known about this hierarchy? Does it collapse? Or is there another name for it in the literature?