  Verifying that a network configuration satisfies a given boolean predicate is
a fundamental problem in distributed computing. Many variations of this problem
have been studied, for example, in the context of proof labeling schemes (PLS),
locally checkable proofs (LCP), and non-deterministic local decision (NLD). In
all of these contexts, verification time is assumed to be constant. Korman,
Kutten and Masuzawa [PODC 2011] presented a proof-labeling scheme for MST, with
poly-logarithmic verification time, and logarithmic memory at each vertex.
  In this paper we introduce the notion of a $t$-PLS, which allows the
verification procedure to run for super-constant time. Our work analyzes the
tradeoffs of $t$-PLS between time, label size, message length, and computation
space. We construct a universal $t$-PLS and prove that it uses the same amount
of total communication as a known one-round universal PLS, and $t$ factor
smaller labels. In addition, we provide a general technique to prove lower
bounds for space-time tradeoffs of $t$-PLS. We use this technique to show an
optimal tradeoff for testing that a network is acyclic (cycle free). Our
optimal $t$-PLS for acyclicity uses label size and computation space $O((\log
n)/t)$. We further describe a recursive $O(\log^* n)$ space verifier for
acyclicity which does not assume previous knowledge of the run-time $t$.
