  Temporal logics have been studied as an approach to the specification of
hyperproperties, resulting in the conception of "hyperlogics". With a few
recent exceptions, the hyperlogics thus far developed can only relate different
traces of a transition system synchronously. However, important information is
contained in the relation between different points in their asynchronous
interaction. To specify such "asynchronous hyperproperties", new trace
quantifier based hyperlogics have been developed. Yet, hyperlogics with trace
quantification cannot express certain requirements that describe the
relationships between all executions of a system. Also, these logics induce
model checking problems (MC) with prohibitively high complexity costs in the
number of quantifier alternations. We study an alternative approach to
asynchronous hyperproperties by introducing a novel foundation of temporal team
semantics. Team semantics is a logical framework that specifies properties of
sets of traces of unbounded size directly, and thus does not have the same
limitation as the quantifier based logics mentioned above. We consider temporal
team logics which employ quantification over so-called "time evaluation
functions" (TEFs) controlling the asynchronous progress of traces instead of
quantification over traces. TEFs constitute a novel approach to defining
expressive logics for hyperproperties where diverse asynchronous interactions
between computations can be formalised and enforced. We show embeddings of
synchronous TeamLTL into our new logics. We show that MC for some TeamCTL
fragment is highly undecidable. We present a translation from TeamCTL* to
Alternating Asynchronous B\"uchi Automata, and obtain decidability results for
the path checking problem and restrictions of MC and SAT. Our translation
constitutes the first approach to team semantics based on automata-theoretic
methods.
