I'm very much aware of Penrose's work, and he's right about almost everything.  But this has not much to do with what I'm trying to make you see.  You are confusing the "entropy of the universe" (which is an ill-defined concept) with the entropy of a sub-system in relation to an "observer", which is entirely well-defined, and to which almost all of what is scientifically claimed about entropy, is about.  
This entropy is defined as a function of the possible cases that the observer needs/wants to consider, and hence limited to the sub-system under study.

The (quantum) entanglement you are talking about is in fact exactly the generator of entropy in a system state: it is the fact of limiting one's attention to a sub-system, while this subsystem entangles with the rest of the universe, that changes the quantum state of the subsystem from a pure state into a mixed state, and while the entropy of a pure state (which implies it is also a *known* state) is zero, a mixed state has a (well-defined) entropy.

I don't know how much you're acquainted with this, I can hardly write out a three-year course on quantum physics and statistical physics here, but essentially:

If you have a sub-system A that is a priori isolated from the environment B ("the rest of the universe"), and you happen to know the exact quantum state of A, then the whole universe is in a special product state |a> |b> ; where |a> is the quantum state of A.  The entropy of A as such, with respect to you, is zero, because you know its micro state perfectly, it is |a>.

Once A interacts unavoidably with the rest of the universe, A gets entangled with it, and there's no specific quantum state of A any more.  However, you can still limit your attention to system A.
The quantum state of the whole universe is now a sum of |a1>|b1> + |a2>|b2> + .... |an> |bn> where we have been running over all possible micro states of A: there are n of them.
The quantum description of just system A now reduces to a density operator, which takes on the form:

| a1 > < a1 | () + |a2 > < a2 | () + ... |an > ).

Here, ( ) are positive real numbers, the norms of the quantum states of "the rest of the universe" that are entangled with our system's quantum state ai.  There are only n such states of the universe that MATTER even though there are infinitely more of course, but they are not solicited by the entanglement with our system.

As such, our density operator consists of n quantum states, with statistical weights given by ( ).  In the worst case, these weights are all equal, which means that our density operator corresponds to "total ignorance of the micro state of A".  Usually, we take extra conditions on the interaction with the rest of the universe, like "energy conserving" or "thermal equilibrium".  This changes the values of ( ), and leads to things like micro-canonical ensemble, canonical ensemble and so on.

In the worst case, we don't know which of the n micro states our system is in (due to entanglement with the rest of the universe), and then its entropy is log_2 (n) in bits, or 1/K_b ln(n) in thermodynamic units.

From an information perspective, if a system can be in N possible states, its entropy is (at most) log_2 (N).

Now, let us compare.  A cup of hot water, at 80 centigrade, 100 grams, say, will contain MORE than the entropy increase from freezing to 80 degrees, right ?

Now, the heat delivered to 100 grams of water to go from 0 to 80 centigrade is 418.6 J/K x 80 = 33488 J.
This heat is transferred at a temperature of less than 80 centigrade, so less than 360 K.   As such, the entropy *increase* is at least 93J/K.

Now, this corresponds to a number of states N equal to e^(6.739 10^24), or to a number of bits equal to 9.7 x 10^24.

Note that I didn't even consider the entropy of icy water, which is not zero.

So my cup of hot water has already an entropy of 10^25 bits.  No computing system on earth masters a memory capacity (*) of 10^25 bits, not even the whole internet.

(*) I take memory capacity as "state space" proxy.