# What is the difference between SIZE(n^k) and P/poly?

What is the difference between $$\text{SIZE}(n^k)$$ and $$\text{P}/\text{poly}$$?

For reference:

• $$\text{SIZE}(n^k)$$ is defined as the class of problems solvable with Boolean circuits (of fan-in two) with $$O(n^k)$$ gates.

• $$\text{P}/\text{poly}$$ is defined as those problems over $$\{0,1\}^*$$ which can be solved by an infinite family of polynomial-size circuits $${C_n}$$.

What is the difference between these classes?

SIZE($$n^2$$) consists only of problems that can be solved by circuit families of size at most $$O(n^2)$$. P/poly contains problems that can be solved by circuit families of size at most $$O(n^3)$$, and those solved by families of size at most $$O(n^4)$$, and so on.
In particular, P/poly = $$\cup_k$$ SIZE($$n^k$$).