Editing Fast Quantum Byzantine Agreement (section)

==Outline==
Here we will sketch the outline of the Fast Quantum Byzantine Agreement protocol by Ben-Or [[Quantum Byzantine Agreement#References|(3)]] that solves Byzantine Agreement using quantum resources. A very nice summary of this protocol is also presented in [[Quantum Byzantine Agreement#References|(1)]].
The main idea of this protocol is for each player to classically send its proposed input bit <math>b_i</math> to every other player in the network and then collaborate to determine what bit is proposed by a majority of honest players. In the case where failed players make this difficult, a 'good-enough' random coin is globally flipped  (using quantum resources, explained below), which is then classically post-processed to reach agreement among the honest parties. Let us make this more precise.

The protocol consists of consecutive rounds. Initially, each player sets a decision bit to its input bit. Then in each round, the players take the following steps:

* Each player transmits its current decision bit to every other player. If a player receives the same bit value from more than 2/3 of the players (including his own), then it sets his decision bit to this majority bit value. Otherwise, that player initiates a Quantum Oblivious Common Coin subroutine with all other players and sets his decision bit to the outcome of this subroutine.

* Then each player sequentially executes two classical subroutines to bias the decision value towards <math>0</math> or <math>1</math>  respectively. These subroutines guarantee that if the non-faulty players are in agreement, then they will terminate and successfully output the correct agreement value.
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'''Quantum Oblivious Common Coin subroutine:''' 
The heart of this protocol comes from the quantum enhanced [[Oblivious Common Coin]]. At the end of this subroutine, each player <math>i</math> outputs a random bit <math>v_i</math>, such that with at least probability <math>p</math> (called the fairness) <math>v_i = x</math> for all players <math>i</math> and all <math>x \in \{0,1\} </math>. Intuitively, this subroutines tosses a common coin, where all players get either <math>0</math> or <math>1</math> with probability at least <math>p</math> each, but there may be executions (which occur with at most probability <math>1-2p</math>) where all players do not get the same output and no common coin is actually tossed. Since the players do not know whether the outcomes are all equal or not, this type of coin tossing is referred to
as oblivious common coin tossing. In particular, using quantum resources, this task can be achieved in constant rounds (in the defined model). The implementation of this subroutine makes use of a weakened
version of [[Verifiable Quantum Secret Sharing]] (VQSS).