Complexity class of sensitivity Let $f:\{0,1\}^n\rightarrow \{0,1\}$ be a Boolean function.

Let $[n]=\{1,2,\dots,n\}$.

If $i\in[n]$, let $\Bbb 1_i$ be length $n$ vector with all $0$s except $1$ at $i$th position.

If $B\subseteq [n]$, then $\Bbb 1_B$ be the length $n$ vector with $1$s only in positions marked by $B$.

If  $i\in[n]$ and $x\in\{0,1\}^n$, let $x^i=x\oplus\Bbb 1_i$ where $\oplus$ is $XOR$ operation.

If  $B\subseteq [n]$ and $x\in\{0,1\}^n$, let $x^{B}=x\oplus\Bbb 1_B$ where $\oplus$ is $XOR$ operation.

Sensitivity of $f$ at input $x$ is $$S_x(f) = |\{i:f(x)\neq f(x^i)\}|$$

Sensitivity of $f$ is $$S(f)=\max_xS_x(f)$$

Block Sensitivity of $f$ at input $x$, $BS_x(f)$ is maximum $r$ such that there is a set of disjoint subsets $\{B_i\}_{i=1}^r$($\forall i\neq j$, $B_i\cap B_j=\emptyset$) such that  $$\forall j\mbox{, }f(x)\neq f(x^{B_j})$$

Block Sensitivity of $f$ is $$BS(f)=\max_xBS_x(f)$$

Given $f$ in $n$-variables, would it be reasonable to ask the complexity class of deciding $S(f)&gt;c$ and $BS(f)&gt;c$ for a fixed $c&gt;0$? I understand that describing a Boolean function takes $2^n$ bits fully.

Does computing $BS(f)$ also take $O(2^{n+\epsilon})$ time as in answers below?
[An alternative answer for $BS(f)$]

There is a paper related to this problem.

Scott Aaronson: Algorithms for Boolean Function Query Properties.
SIAM J. Comput. 32(5): 1140-1157 (2003)

The paper gives an $O(N^{\log_2 5} \log N)$ algorithm for computing $BS(f)$ if $f$ is given by
a truth table of size $N=2^n$. Thus, even if $c$ is not fixed, deciding $BS(f) &gt; c$
is in P.