Does uncertainty principle imply that the past history of universe is also undetermined? I read the page here:
Is the Uncertainty Principle valid for information about the past?, but I am still somewhat confused.
If you measure the momentum/position (with uncertainty) of a particle, what could you infer mathematically about its past? Its past momentum/position? Or rather some wave function as there could be infinite different systems with different possible wave functions that are collapsed to generate the measured result?
 A: No, the uncertainty principle is a statement about the localization of a quantum state when represented in position vs. momentum space (more generally, the eigenbases of two non-commuting observables). So long as no measurements take place, the quantum state can be evolved completely deterministically forward and backward in time if you know the Hamiltonian. This is what physicists mean when they say quantum mechanics is deterministic.
Any state, even one translated forward or backward in time using the Hamiltonian, will obey the uncertainty principle: namely the spread of the state in position space must obey a certain inequality in union with the spread of the state in momentum space. This does have some net effect on the outcome of measurements of that quantum state, but that is not the same thing as the state itself. 
The state itself evolves determanistically forward or backward until a measurement occurs. Of course, measurements on the state like "what's your position" or "what's your momentum" will also obey the uncertainty principle, because the outcome of measurements (the part of quantum mechanics that is random) are probabilistically determined by the amplitude of the wavefunction over the eigenspace of the measurement operator. That is to say, the uncertainty principle is not related to randomness of quantum mechanics in any way other than requiring that the state will always have some spread over the eigenbases of non-commuting operators. 
A: If you happened to know the quantum state of the universe as well as the Hamiltonian, then you could use the Schrodinger equation to know the quantum state at both past and future (!) times.
However, that quantum state would still not have both well-defined position and momentum, for instance.
A: The uncertainty principle puts a limit on what can be known, not on what can exist. It is possible that nature is deterministic but our description of it it is probabilistic due to our incomplete knowledge. So the answer to your question is "no". This is also true about the future. Heisenberg's principle does not imply that the future is undetermined, only that we cannot predict it with certainty.
In principle you could infer the past of a particle by performing successive position measurements. Two such measurements, together with an accurate timing of them, allow you to also calculate the momentum of the particle. But this is only possible for the time before the first measurement and the second one, because each measurement alters the momentum. You also need to make some assumptions about how the particle moves, say in straight lines. 
