A Quora question asks: Why are conservation laws related to conservation while systems evolve in time, while there are no equivalent principia that applies to conservation in space? Does this fact constitute an ontologic distinction between time and space?
The distinction between space and time (regardless of whether or not you fancy it up with the silly word “ontologic”) is that from our point of view time is the direction in event-space for which we have memory of one half-space but not the other, and our experience of more recent compared to more distant memories defines for us what it means for systems to evolve. So time for any observer is by definition the direction in which systems evolve for that observer (and this is true regardless of what laws, conservation or otherwise, we use to predict that evolution).
Before the 20th century it was generally assumed that the time direction was the same for all observers, but we now know that it depends to some extent on the observer. Fortunately the evolution of physical systems can be summarized in laws which appear to have the same form for all observers, but the fact that some of these are conservation laws is quite irrelevant to the distinction between time and space.
Some systems do have the property that their structure in one spatial direction can be inferred from knowledge of that in the opposite direction, and in some cases there are quantities that are the same for all values of the relevant coordinate. We don’t normally refer to changes with respect to a spacelike parameter as “evolution”, and although quantities that are independent of position might well be said to be “conserved over space” we generally understand the unmodified word “conserved” to refer to conservation over time.