  We determine the complexity of second-order HyperLTL satisfiability,
finite-state satisfiability, and model-checking: All three are as hard as truth
in third-order arithmetic.
  We also consider two fragments of second-order HyperLTL that have been
introduced with the aim to facilitate effective model-checking by restricting
the sets one can quantify over. The first one restricts second-order
quantification to smallest/largest sets that satisfy a guard while the second
one restricts second-order quantification further to least fixed points of
(first-order) HyperLTL definable functions.
  The first fragment is still as hard as truth in third-order arithmetic while
satisfiability for the second one is $\Sigma_1^1$-complete, i.e., only as hard
as (first-order) HyperLTL and therefore much less complex. Finally,
finite-state satisfiability and model-checking are in $\Sigma_2^2$ and
$\Sigma_1^1$-hard, and thus also less complex than model-checking full
second-order HyperLTL.
