Given a hash function $H:\{0,1\}^*\rightarrow\{0,1\}^n$, a difficulty $d\in\mathbb{N}$, and data $D\in\{0,1\}^*$, the framework of the Hashcash/Bitcoin Proof-of-Work entails finding a nonce $c$ such that $H(c\Vert D)$ starts with $d$ zero’s. That is to say, Hashcash/Bitcoin Proofs-of-Work appear to find a preimage $c\Vert D$ that partially inverts a hash $H$. Depending on $d$ and the size of the nonce, it may be impossible to find a nonce $c$ without having to change the data $D$. Currently (Aug. 2017) with Bitcoin, the target difficulty $d$ is, I think, 17 consecutive hexadecimal 0's.
Given a set $S\subseteq\{0,1\}^m$ and a number $K$, the Goldwasser-Sipser Set Lower Bound Protocol is an Arthur-Merlin protocol for Merlin the prover to show Arthur the verifier that $|S|\geq K$. Letting $k$ be a number such that $2^{k-2}\leq K\leq 2^{k-1}$, Arthur chooses a random hash $H$ from pairwise independent hash function collection, along with a random $y\in\{0,1\}^k$. Merlin’s goal is to find an $x\in S$ such that $H(x)=y$. That is to say, Merlin’s goal in the Goldwasser-Sipser Set Lower Bound Protocol is to find a preimage $x$ that inverts a hash function $H$. The larger the set $S$, the easier it may be to find a preimage. Indeed, if $S\leq \frac{K}{2}$, then Merlin has a much smaller chance of finding a preimage than if $S\geq K$. Thus, Merlin may provide evidence to Arthur that $S$ is large. This may be used, e.g., to show that two graphs are not isomorphic.
My question is: can we reinterpret the Hashcash/Bitcoin Proof-of-Work as a Set Lower Bound? In other words, can we consider finding a nonce $c$ in the Bitcoin Proof-of-Work as similar to finding a preimage $x$ in the Goldwasser-Sipser Set Lower Bound, which may show that some (arbitrary) set is large?
Or is the similarity between the two not very strong?
EDIT
Following up on D.W.'s answer, I wonder if $D$ could be considered as the input to a test of membership of $x\in S$. For example, $S$ could be the set of all permutations of two graphs $G_1$ and $G_2$, $D$ might "encode" the two graphs $G_1$ and $G_2$, and $x$ might "encode" a permutation $\pi$ of one of $G_1$ or $G_2$. The fact that a preimage $x$ was found might indicate that $G_1$ is not isomorphic to $G_2$, because it indicates $S$ is large.
However, I think there may be no easy way to tell what the "encoding" is. So I wonder if a Proof-of-Work can, by design, be based on some interesting problem in $\mathsf{coNP}$, by having the encodings of the problem specified a-priori.
For example, the adjacency matrix of the two graphs $G_1$ and $G_2$ may be encoded in $D$. The miner may engage in looking for another graph $G_3$ that is a permutation $\pi$ of one of $G_1$ or $G_2$, such that the hash $H_2$ is $0$. That is, the miner may treat $G_3=\pi(G_1)$ or $G_3=\pi(G_2)$ as $x$ (i.e. the "nonce" $c$.) The miner may "succeed" if $H(\pi(G_1) \Vert G_1 \Vert G_2)$ begins with $d$ zeroes, or $H(\pi(G_2) \Vert G_1 \Vert G_2)$ begins with $d$ zeroes, thus indicating that the number of permutations of $G_1$ and $G_2$ together is "large" and not "small."