  We study the power of Arthur-Merlin probabilistic proof systems in the data
stream model. We show a canonical $\mathcal{AM}$ streaming algorithm for a wide
class of data stream problems. The algorithm offers a tradeoff between the
length of the proof and the space complexity that is needed to verify it.
  As an application, we give an $\mathcal{AM}$ streaming algorithm for the
\emph{Distinct Elements} problem. Given a data stream of length $m$ over
alphabet of size $n$, the algorithm uses $\tilde O(s)$ space and a proof of
size $\tilde O(w)$, for every $s,w$ such that $s \cdot w \ge n$ (where $\tilde
O$ hides a $\polylog(m,n)$ factor). We also prove a lower bound, showing that
every $\mathcal{MA}$ streaming algorithm for the \emph{Distinct Elements}
problem that uses $s$ bits of space and a proof of size $w$, satisfies $s \cdot
w = \Omega(n)$.
  As a part of the proof of the lower bound for the \emph{Distinct Elements}
problem, we show a new lower bound of $\Omega(\sqrt n)$ on the $\mathcal{MA}$
communication complexity of the \emph{Gap Hamming Distance} problem, and prove
its tightness.
