The quantum eraser is a clever experimental design, in which first a quantum system, prepared to be in a superposition of states corresponding to various values of an observable O, undergoes a “measurement” by getting entangled with another microscopic system playing the role of “measuring apparatus,” so that the value of O gets correlated with the value of an observable M of the “apparatus.” Then this correlation gets destroyed by performing on the “apparatus” the measurement of an observable incompatible with M.
A concrete example is a Mach-Zehnder interferometer in which a photon can travel along two different paths and interfere with itself. If we place on each path a device that generates a secondary photon when the primary photon crosses it, the detection of the secondary photons provides which path information about the primary photon and the interference pattern disappears. But the trajectories of the two secondary photons can be mixed in such a way that information about which is which is lost: consequently information about the primary photon’s path is also lost, and the interference pattern reappears.
The experiment shows that persistence of information is the crucial factor allowing the replacement of the entangled superposition that is the outcome of a measurement process with a classical alternative (the so-called “collapse of the state vector”). The replacement holds, in other words the outcome of a measurement process can be seen in objective either/or terms, so long as some kind of record or trace of the measurement results persist. When all that information is erased, the replacement is no longer possible and only the full entangled superposition correctly predicts the results of future observations. The collapse of the state vector is therefore merely an apparent phenomenon. The outcome of a measurement process is always an entangled superposition. But for predicting the outcome of an observation that conserves information about the measurement process, the entangled superposition is exactly equivalent to a classical alternative. 
The quantum eraser can be performed as a delayed choice experiment. E.g., in the case of the Mach-Zehnder interferometer mentioned above, the mixing of the trajectories of the secondary photons can be applied when the primary photon is already on its way and even when it has already interacted with the devices generating the secondary photons, so that the decision to move along one path or the other, or both at once, should have been already taken.
This aspect of the experiment looks like a retroactive effect in time, and it has been the object of much discussion. A particularly striking version of delayed choice are some thought-experiments proposed by John Archibald Wheeler (1911-2008) that stretch the time delay to cosmic proportions.
Here is one such experiment (fig. 4.5.) Light coming towards the Earth from a distant star is deflected by a massive celestial object, e.g. a galaxy or a giant black hole, acting as a gravitational lens. Then (looking at the incoming photons trajectories in only two dimensions for simplicity), just like in the case of the quantum eraser, each photon travels along two paths around obstacle, paths that finally converge in our lab here on Earth.
Now the incoming beams can be brought together, producing an interference pattern, or can be separately analysed, revealing only the photons coming from one or the other direction. So our choice of experimental apparatus seems to force the photon to choose between its wave nature, passing simultaneously on both sides of the obstacle, and its particle nature, passing either on one side or on the other. This in itself is not surprising: it is a well-known feature of quantum physics that incompatible experimental arrangements bring to light complementary aspects of the observed phenomenon (e.g., particle or wave behavior). What is rather extraordinary, though, is that the galaxy causing the deflection of the light beams may be millions of light-years away, so that a photon passing just on one side of it or on both sides at once is an event that supposedly took place millions of years ago. It would seem that our choice of experimental setup today influences the past millions of years ago.
Wheeler understood this backward influence in time in the context of his participatory universe. He said:
The thing that causes people to argue about when and how the photon learns that the experimental apparatus is in a certain configuration and then changes from wave to particle to fit the demands of the experiment’s configuration is the assumption that a photon had some physical form before the astronomers observed it. Either it was a wave or a particle; either it went both ways around the galaxy or only one way. Actually, quantum phenomena are neither waves nor particles but are intrinsically undefined until the moment they are measured [emphasis mine]. In a sense, the British philosopher Bishop Berkeley was right when he asserted two centuries ago “to be is to be perceived.”
We are participators in bringing into being not only the near and here but the far away and long ago. We are in this sense participators in bringing about something of the universe in the distant past…”
The persistence of information approach fits well with the notion of a participatory universe, but this participation does not need to be understood in the sense that we “bring about something of the universe in the distant past”. The apparent retroactive effect in time is merely a reflection of the inadequacy of our representations of matter (which are imbued with classical prejudice). In particular, the notions of particle and wave are both inadequate: they are just our last attempt to extend to the micro world familiar notions of our macro world.
The only consistent description of the micro world is in terms of the quantum state of the system together with the laws that give the probabilities of specific results for specific measurements that can be performed on the system. The photon, interacting with the far away galaxy, ends up in a quantum state which is a superposition of travel on both sides of the galaxy. Whether in our observations on Earth we see that superposition (in the form of an interference pattern) or we see a classical alternative (separate counts of photons in each beam) depends on whether our specific setup is such as to erase or keep which path information. E.g., if we catch each beam in a properly oriented telescope, which path information is conserved and we have no interference. If we have the two beams converge on a photographic plate, which path information is lost and we have an interference pattern.
Our choice of the apparatus setup in the cosmic delayed choice experiment therefore does not change anything in the distant past. What happened in the distant past is the creation of an entangled state in which the photon is travelling on both sides of the galaxy. What happens here now is that we can choose to perform on that state an observation conserving or destroying which path information. In the first case the entangled superposition is equivalent to a classical alternative. The photon appears to have traveled either one path or the other: good particle behavior. In the second case there is no such equivalence, and the entangled superposition is the only correct description. The photon appears to have traveled along both paths at once: good wave behavior.
But there is no retroactive change, since all results, whether they do or do not conserve which path information, can be derived from the full entangled superposition representing the present state of the system. Our choice of the experimental setup determines what we see today: it does not affect what happened millions of years ago to the photon and the galaxy. Shantena A. Sabbadini, Persistence of Information in the Quantum Measurement Problem, Physics Essays, March 2006, Vol. 19 No. 1, pp. 135-150.  Scientific American, July 1992, p. 75.  “The Anthropic Universe”, radio interview in Science Show, 18 February 2006.