  In a distributed coin-flipping protocol, Blum [ACM Transactions on Computer
Systems '83], the parties try to output a common (close to) uniform bit, even
when some adversarially chosen parties try to bias the common output. In an
adaptively secure full-information coin flip, Ben-Or and Linial [FOCS '85], the
parties communicate over a broadcast channel and a computationally unbounded
adversary can choose which parties to corrupt along the protocol execution.
Ben-Or and Linial proved that the $n$-party majority protocol is resilient to
$O(\sqrt{n})$ corruptions (ignoring poly-logarithmic factors), and conjectured
this is a tight upper bound for any $n$-party protocol (of any round
complexity). Their conjecture was proved to be correct for single-turn (each
party sends a single message) single-bit (a message is one bit) protocols
Lichtenstein, Linial and Saks [Combinatorica '89], symmetric protocols
Goldwasser, Tauman Kalai and Park [ICALP '15], and recently for (arbitrary
message length) single-turn protocols Tauman Kalai, Komargodski and Raz [DISC
'18]. Yet, the question for many-turn protocols was left completely open.
  In this work we close the above gap, proving that no $n$-party protocol (of
any round complexity) is resilient to $\omega(\sqrt{n})$ (adaptive)
corruptions.
