  We investigate opinion dynamics in a fully-connected system, consisting of
$n$ identical and anonymous agents, where one of the opinions (which is called
correct) represents a piece of information to disseminate. In more detail, one
source agent initially holds the correct opinion and remains with this opinion
throughout the execution. The goal for non-source agents is to quickly agree on
this correct opinion, and do that robustly, i.e., from any initial
configuration. The system evolves in rounds. In each round, one agent chosen
uniformly at random is activated: unless it is the source, the agent pulls the
opinions of $\ell$ random agents and then updates its opinion according to some
rule. We consider a restricted setting, in which agents have no memory and they
only revise their opinions on the basis of those of the agents they currently
sample. As restricted as it is, this setting encompasses very popular opinion
dynamics, such as the voter model and best-of-$k$ majority rules.
  Qualitatively speaking, we show that lack of memory prevents efficient
convergence. Specifically, we prove that no dynamics can achieve correct
convergence in an expected number of steps that is sub-quadratic in $n$, even
under a strong version of the model in which activated agents have complete
access to the current configuration of the entire system, i.e., the case
$\ell=n$. Conversely, we prove that the simple voter model (in which $\ell=1$)
correctly solves the problem, while almost matching the aforementioned lower
bound.
  These results suggest that, in contrast to symmetric consensus problems (that
do not involve a notion of correct opinion), fast convergence on the correct
opinion using stochastic opinion dynamics may indeed require the use of memory.
This insight may reflect on natural information dissemination processes that
rely on a few knowledgeable individuals.
