In a deterministic universe, if you’re willing to accept that the past state of the universe explains the present, you’ve also got to accept that the future state of the universe explains the present, in the same sense. That’s just a property of the laws of physics: the data at any Cauchy surface fixes the complete evolution. So for what happens in the present, you could appeal to the laws of physics and the conditions at the Big Bang; or the laws of physics and the conditions one second ago; or the laws of physics and the conditions next Friday: they’re equivalent ways of stating the same thing.
Not in a perfectly deterministic world, no (which we were positing in the part of the discussion you’re responding to). There, the laws of physics are invertible transformations of the state at any given point in time—that is, if the state now is given by s, and at time t by U_t(s), then there exists U_t^{-1} such that U_t^{-1}(U_t(s))=s. This is, for instance, true in quantum mechanics without collapse (e.g. the many-worlds interpretation). But if you introduce a collapse, then on every collapse event, you lose information, and the present state won’t suffice to reconstruct the prior state.