As has been said by others...
More information is needed, specifically what are the speaker's rated sensitivity. If the sensitivity is low (below 87dB) then you'll need more amp power than speakers with higher sensitivity (above 90dB). Digging deeper, you'll need to know the speakers impedance across the audible frequency range, because speakers have impedance peaks at different frequencies. This is where an amp's current rating comes into play and how well it can provide power through high impedance loads - which in many cases are spontaneous frequency-based loads that will affect sound quality and overall audible response (high-pitch cymbal crash causes a spike in impedance and suddenly the bass is affected because the amp can't provide the power).
In other words, to best answer the question, the make and model of the amp and the speakers is needed to offer the best answer to your question.
Another point is about the description of 1/4 volume and 1/2 volume. This is meaningless because most integrated amps and preamps do not have linear volume control - in other words, 1/2 volume knob adjustment is not necessarily 1/2 volume. I'd completely disregard volume knob fractional increments with respect to actual SPL.
More power is almost always better for sound quality (and loudness) because of the increased head room for instantaneous dynamics. But what matters even more is quality amplification - look for amps that can deliver more power into ever decreasing impedances down to 2 ohms.
Example: The NAD C388 integrated amp is rated:
8 Ohm 250W
4 Ohm 350W
2 Ohm 400W
Please note this example is not high-end by 'audiophile standards' - as a matter of fact it would likely be considered mid-Fi, and is so loaded with features that many are gasping right now that I even mentioned what might be considered a borderline 'lifestyle' product. Its essentially a extremely supped-up Bluesound PowerNode with display, phono stage, dual sub outs, line level outs, MDC card expandability, etc.
Whenever I'm shopping for amps, the first must-have is 2ohm stable; otherwise, its obsolete soon as those demanding (low-impedance) speakers catch your eye.
