Circuit complexity has connections to many questions in complexity theory. For a couple examples, Ryan Williams shared some in a recent talk and Section 3 of these notes gives simple relations to $\mathbf{P}$ vs. $\mathbf{NP}$ and $\mathbf{P}$ vs. $\mathbf{BPP}$. More can be found in the textbook Boolean Function Complexity. On the other hand, the Unique Games Conjecture has been linked to a wide range of computational problems and areas of applied mathematics; see e.g. this talk or this survey of Khot.
What I'm wondering is, are there strong connections known between circuit complexity and hardness of approximation, specifically the Unique Games Conjecture? If so, what are they?
