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We consider the algorithmic problem of community detection in networks. Givenan undirected friendship graph $G=\left(V,E\right)$, a subset $S\subseteq V$ isan $\left(\alpha,\beta\right)$-community if:  * Every member of the community is friends with an $\alpha$-fraction of thecommunity;  * Every non-member is friends with at most a $\beta$-fraction of thecommunity.  Arora et al [AGSS12] gave a quasi-polynomial time algorithm for enumeratingall the $\left(\alpha,\beta\right)$-communities for any constants$\alpha>\beta$.  Here, we prove that, assuming the Exponential Time Hypothesis (ETH),quasi-polynomial time is in fact necessary - and even for a much weakerapproximation desideratum. Namely, distinguishing between:  * $G$ contains an $\left(1,o\left(1\right)\right)$-community; and  * $G$ does not contain an$\left(\beta+o\left(1\right),\beta\right)$-community for any$\beta\in\left[0,1\right]$.  We also prove that counting the number of$\left(1,o\left(1\right)\right)$-communities requires quasi-polynomial timeassuming the weaker #ETH.

Detecting communities is hard, and counting them is even harder