Was the Universe's entropy equal to zero at the Big Bang? Is zero-entropy state unique? It is postulated by many cosmologists that at the Big Bang time the universe was in an unusual low entropy state.
Does this claim specifically mean that the entropy of the initial universe was zero?
Is zero-entropy state unique for given physical laws?
Is it possible that entropy was growing always so that only difference in entropy has physical meaning rather than absolute value? Was there ever negative entropy state?
 A: Whether entropy was zero at the Big Bang or not is very much an open question of physics, in big part due to the fact that we do not yet have a good enough understanding of physics at high energies and high gravitational fields.
But for the zero entropy state this is a bit easier to answer and the answer does depend on laws of physics. Zero entropy state basically depends on how many completely distinguishable states the laws of physics allow. The universe is in a zero entropy state precisely when it is in a single state and it can be known which state it is in. In many situations there are infinitely many different zero entropy states. So the zero entropy state at the beginning of the universe is unique if and only if the laws of physics at that time require that there is a single state in which the universe can be found. Whether they do require that or not is a very big question in physics which everybody would like to know the answer to.
A: The correct answer is we just don't know and never will . This is because all known Physical laws breakdown at about $10^{-42}$ seconds after the  Big Bang  or if you will Initial Inflation . Quantum calculations will not render an answer , and all hypotheses are not testable thus not provable . This is simply because we cannot roll back the Universe to it's Origin , but only to $10^{-42}$ seconds after it started it's inflation .At this point Science becomes Philosophical . We have absolutely no idea of what Physical Laws drove the Universe into the existence we now observe . Not even Hawking can do better than guess . 
A: I understand that there was zero entropy or at least low entropy that became self agitated to bring about the big bang. The closest example to zero or very low entropy is a solution of Dropleons with quasiparicles structures in a liquid state that are self propagating. This is  based on the initial conditions of the solution containing super conducting and colloidal suspended super conductive quasiparticles. As the quasiparticles disappear in a flash of time they give off energy and this bombards the super conducting quasiparticls,
Based on the Shannon Curve there is an optimum combination of Quantum Droplets in solution along with the other quasiparticles, all on a quantum level of size. This optimum state of mixture that has also been exposed to light frequencies responsible for creating Excitons and Dropletons results in zero or very low entropy and very high energy.
But in this state of zero or near zero state of entropy nothing is really happening. Since this solution is man made it will take an intentional input to cause agitation in the solution that still constantly tries to maintain zero entropy. The result is what some scientists call artificial intelligence but I consider it to be Natural. I call this device LISA-T, an abbreviation for Living Intelligence Self-Aware Technology. The agitation is created with the correct application of magnetic influx.
Dave Wheeler 
A: It depends what you mean by big-bang. I consider the big-bang to begin with inflation, not with a singularity, so that the starting point is the inflating universe, making no hypotheses about what came before (if the question even makes sense).
The inflating initial starting state is for all intents and purposes, a perfect deSitter state which is adiabatically growing as the inflaton slides down the potential. At the end of inflation, when the inflaton starts shaking non-thermally, the state is no longer unique, but the semiclassical description of the initial state is by a thermal state inside a deSitter horizon.
The natural entropy to associate with this state is the area of the cosmological horizon in Planck units, and this entropy is far from zero. But it is infinitesimally small compared to the maximum entropy we could squeeze into the universe today, given that the cosmological horizon has grown so much, but past the end of inflation, the growth has been out-of-equilibrium.
So the entropy of the initial state of the universe is about the square of the de-Sitter radius at the end of inflation. I don't know a precise number, but suppose it's a deSitter temperature of $10^{14}$ GeV, that's about a million planck lengths of radius, so a dimensionless entropy of order $10^{12}$. Compare with $10^{135}$, which is the maximum entropy you can squeeze in the modern cosmological horizon, and you can see how low-entropy the initial state was, despite being in thermal equilibrium at the time.
This explanation of the low-entropy initial conditions requires you to consider a single horizon-volume as all there is, and this is the holographic view of inflation promoted by Banks, Fischler, Shenker and Susskind. It was suggested to be the reason for the low-entropy initial condition by Davies in the early 1980s, but it is still not accepted by the astrophysical community, for reasons that I wouldn't be able to properly explain, because I think they are ridiculous.
