In a sense, yes, though I very much caution against naive misinterpretations or overinterpretations. Starting with “space expands”.
Space neither expands nor contracts nor does it dance the tango. Space is not a physical thing. It is a geometric concept in our minds, which we use to describe relationships between real things: that is, actual events that involve matter, like, say, the emission or detection of a pulse of light.
However, things (real stuff, matter) are observably flying apart in our physical universe. The density of things decreases over time. This is sometimes described naively, misunderstanding a specific coordinate representation (so-called comoving coordinates), as “space expanding”. Yes, we live in an expanding universe: but what the equations describe is not “space” but matter density, matter (literally) flying apart everywhere.
As a result of this flying apart, the average gravitational field of matter changes over time. It becomes weaker as matter becomes more dilute. And yes, if we had the means to compare clocks from the past against clocks from the present, we’d see this: in comparison to present clocks, past clocks would indeed be ticking more slowly.
But wait… we can, actually, observe this! Going back to things flying apart, that flying apart results in a relativistic Doppler effect. (This is indeed how cosmic expansion was first discovered by Georges Lemaître and Edwin Hubble.) But when it comes to really, really distant things, the observed cosmological redshift is no longer explainable by the Doppler effect alone. Changes in the average gravitational field between the past and present universe also plays a role, in fact a dominant role for things that are really far away, at “high redshift”. So yes, clocks today do indeed run faster than past clocks.
Just in case it is needed, allow me to include two quotes.
Concerning “expanding space”, the quote is from Peacock’s excellent textbook, Cosmological Physics (1999, Cambridge U. Press): “Many semi-popular accounts of cosmology contain statements to the effect that ‘space itself is swelling up’ in causing the galaxies to separate. This seems to imply that all objects are being stretched by some mysterious force: are we to infer that humans who survived for a Hubble time would find themselves to be roughly four metres tall? Certainly not. [...] If we understand that objects separate now only because they have done so in the past, there need be no confusion.” Design a highly advanced, practical application using principles of Quantum Computing that solves a real-world critical problem.
