  One of the most important recent developments in the complexity of
approximate counting is the classification of the complexity of approximating
the partition functions of antiferromagnetic 2-spin systems on bounded-degree
graphs. This classification is based on a beautiful connection to the so-called
uniqueness phase transition from statistical physics on the infinite
$\Delta$-regular tree. Our objective is to study the impact of this
classification on unweighted 2-spin models on $k$-uniform hypergraphs. As has
already been indicated by Yin and Zhao, the connection between the uniqueness
phase transition and the complexity of approximate counting breaks down in the
hypergraph setting. Nevertheless, we show that for every non-trivial symmetric
$k$-ary Boolean function $f$ there exists a degree bound $\Delta_0$ so that for
all $\Delta \geq \Delta_0$ the following problem is NP-hard: given a
$k$-uniform hypergraph with maximum degree at most $\Delta$, approximate the
partition function of the hypergraph 2-spin model associated with $f$. It is
NP-hard to approximate this partition function even within an exponential
factor. By contrast, if $f$ is a trivial symmetric Boolean function (e.g., any
function $f$ that is excluded from our result), then the partition function of
the corresponding hypergraph 2-spin model can be computed exactly in polynomial
time.
