The only physical theory we currently possess that is capable of describing cosmology is general relativity (GR). Although we can to some extent interface quantum mechanics with GR, they are basically incompatible, and we don't know how to reconcile them. Therefore it's natural to answer this question within the framework of classical GR.
The way classical GR expresses the notion of cause and effect is as follows. You start with a Cauchy surface, which is a spacelike surface such that every timelike curve intersects it exactly once. (This is the equivalent of fixing a time t in newtonian mechanics.) Then given the initial conditions on this surface, GR allows us to extrapolate forward or backward in time. The extrapolation can fail if you hit a singularity, or if you have closed, timelike curves ("time machines"). A spacetime is called globally hyperbolic if the extrapolation always works. 651
A globally hyperbolic spacetime is one in which cause and effect hold.
An example of a spacetime in which cause and effect fails is one that contains a timelike singularity. Such a singularity can absorb or emit arbitrary energy and information. Standard big bang and black hole models contain only spacelike singularities, and therefore they are globally hyperbolic and allow a sensible notion of cause and effect.
In GR, a singularity is something that is missing from the spacetime manifold. Therefore the big bang singularity cannot be used as a Cauchy surface or as part of a Cauchy surface.
In cosmological models, it's useful to define a time $t$ which is the time on a clock that has been at rest relative to the Hubble flow ever since the big bang. Then if you pick any $t>0$, it defines a valid Cauchy surface (although most Cauchy surfaces are not of this special type).
Therefore the answer to your question is sort of the opposite of what you imagined. The big bang is uniquely ill suited to stating a set of initial conditions for the universe. Any $t>0$ works fine, but there is no $t=0$, since the big bang singularity isn't even part of the spacetime.
An answer by <PERSON> says:
All the energy we have now was determined at the Big Bang
This is wrong, both for the reasons described above and because GR doesn't have global conservation of energy.
Also:
So even though the present events depend on the original BB, the path to them is not reversible.
The <PERSON> equation has perfect time-reversal symmetry. Other answers have given more competent descriptions of the quantum aspects of this question. 43