The arrows indicate secondary dominant chords, as evident in mm3-4 where V7/ii leads to ii, and in mm7-8 where V7/IV leads to IV.
Given that, we have:
E7 = V7/V, because of the arrow leading to a V7 chord. E7 is the dominant chord of A, so the chord following E7 is A7.
Similarly, since A7 is indicated as a V chord, and a I(maj7) chord follows, the latter chord is Dmaj7. The excerpt is in D Major
By the same logic, we know that E7 is the target of an applied dominant chord, which would be B7. Thus, the chord immediately preceding the E7 is B7.
Extending this across the entire chord sequence, we have:
F#7 B7 | E7 A7 | Dmaj7 B7 | E-7 A7 | V7/VI V7/II | V7/V V7 | Imaj7 V7/II | II-7 V7 | Dmaj7 B7 | E7 A7 | F#-7 D7 | Gmaj7 A7 | Dmaj7 || Imaj7 V7/II | V7/V V7 | III-7 V7/IV | IVmaj7 V7 | I || Note 1: My interpretation of the exercise is that the first chord, for example, is V7/VI. However, it also could be called V7/V7/V7/V. Note 2: That we're in D major and not D minor is confirmed by the qualities of the II-7, III-7, and IVmaj7 chords. Note 3: The (3) above the first chord is indicating that the chord is built on the 3rd scale degree of the home key.
This quiz probably is from a book chapter where the chapter surely must make clear what the point of the quiz is.
But just looking at the root movements, almost everything is roots by descending fifth:
V7/II II-7 V7 Imaj7 and harmonic sequence
E7 V7 Imaj7 treated as all descending fifths would be
V7/V V7 Imaj7 or
E7 A7 Dmaj7.
fill in the remainder of the chord symbols.
But something worth mentioning is another aspect about harmony, root progressions, and the clues the quiz gives. In measure 2 beat 1 you have the
E7 which we will say is in
D and labeled
V7/V, but then later in measure 4 beat 1 have have
II-7 which assuming no key change will be
Emin7. The quality of the chord changes from a dominant seventh to a minor seventh, but the root progression didn't change. It's easier to see the unchanged roots using letters: mm. 2-3
E A D and mm. 3-4
E A D. That kind of treatment of chord quality in a descending fifth progression is common. You can choose chord qualities pretty feely.
So, the chord at m.6 beat 1 is a toss up. It could be
V7/V. Same goes for the opening two measures. You could choose either dominant sevenths or minor sevenths. IMO, either would be justified with the given treatment of of root
E as either